id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-22700 | Characterization of Perfectly T4 Space | A topological space $T = \struct {S, \tau}$ is perfectly $T_4$ {{iff}} every closed set is a zero set. | === Sufficient Condition ===
Let $T$ be a perfectly $T_4$ space.
By definition, every closed set in $T$ is a $G_\delta$ set.
Moreover, since $T$ is also a $T_4$ space, by Zero Sets are Exactly the Closed G-Delta Sets in T4 Space, every closed $G_\delta$ set is a zero set.
Hence, every closed set in $T$ is a zero set. | A [[Definition:Topological Space|topological space]] $T = \struct {S, \tau}$ is [[Definition:Perfectly T4 Space|perfectly $T_4$]] {{iff}} every [[Definition:Closed Set (Topology)|closed set]] is a [[Definition:Zero Set|zero set]]. | === Sufficient Condition ===
Let $T$ be a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]].
By definition, every [[Definition:Closed Set (Topology)|closed set]] in $T$ is a [[Definition:G-Delta Set|$G_\delta$ set]].
Moreover, since $T$ is also a [[Definition:T4 Space|$T_4$ space]], by [[Zero Sets are Exactly ... | Characterization of Perfectly T4 Space | https://proofwiki.org/wiki/Characterization_of_Perfectly_T4_Space | https://proofwiki.org/wiki/Characterization_of_Perfectly_T4_Space | [
"Perfectly T4 Spaces",
"Zero Sets"
] | [
"Definition:Topological Space",
"Definition:Perfectly T4 Space",
"Definition:Closed Set/Topology",
"Definition:Zero Set"
] | [
"Definition:Perfectly T4 Space",
"Definition:Closed Set/Topology",
"Definition:G-Delta Set",
"Definition:T4 Space",
"Zero Sets are Exactly the Closed G-Delta Sets in T4 Space",
"Definition:Closed Set/Topology",
"Definition:G-Delta Set",
"Definition:Zero Set",
"Definition:Closed Set/Topology",
"Def... |
proofwiki-22701 | Norm Induced by Projections Associated with Schauder Basis is Norm | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\GF$.
Let $E = \sequence {e_n}_{n \mathop \in \N}$ be a Schauder basis for $X$.
Let $\sequence {S_n}_{n \mathop \in \N}$ be the projections associated with $\sequence {e_n}_{n \mathop \in \N}$.
Define $\norm {\, \cdot \,}_E : X... | === $\norm {\, \cdot \,}_E$ is well-defined ===
We need to show that:
:$\ds \sup_{n \mathop \in \N} \norm {\map {S_n} x} < \infty$ for each $x \in X$.
From Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology and Characterization of Convergence in Strong Operator Topology we have:... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $E = \sequence {e_n}_{n \mathop \in \N}$ be a [[Definition:Schauder Basis|Schauder basis]] for $X$.
Let $\sequence {S_n}_{n \mathop \in \N}$ be the [[Definition:Projections Associated wi... | === $\norm {\, \cdot \,}_E$ is well-defined ===
We need to show that:
:$\ds \sup_{n \mathop \in \N} \norm {\map {S_n} x} < \infty$ for each $x \in X$.
From [[Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology]] and [[Characterization of Convergence in Strong Operator Topology]... | Norm Induced by Projections Associated with Schauder Basis is Norm | https://proofwiki.org/wiki/Norm_Induced_by_Projections_Associated_with_Schauder_Basis_is_Norm | https://proofwiki.org/wiki/Norm_Induced_by_Projections_Associated_with_Schauder_Basis_is_Norm | [
"Schauder Bases"
] | [
"Definition:Banach Space",
"Definition:Schauder Basis",
"Definition:Projections Associated with Schauder Basis",
"Definition:Norm/Vector Space"
] | [
"Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology",
"Characterization of Convergence in Strong Operator Topology",
"Modulus of Limit/Normed Vector Space",
"Definition:Convergent Sequence"
] |
proofwiki-22702 | Perfectly T4 Property is Hereditary | Let $T = \struct{S, \tau}$ be a perfectly $T_4$ space.
Then every subspace of $T$ is also perfectly $T_4$.
That is, the property of being perfectly $T_4$ is hereditary. | {{Recall|Perfectly T4 Space|perfectly $T_4$ space}}
{{:Definition:Perfectly T4 Space}}
Let $H \subseteq S$ be a subspace of $T$, with the subspace topology $\tau_H$.
Let $F \subseteq H$ be a closed set in $H$.
Then, by Closed Set in Topological Subspace there exists a closed set $F_0$ of $T$ such that $F = F_0 \cap H$.... | Let $T = \struct{S, \tau}$ be a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]].
Then every [[Definition:Topological Subspace|subspace]] of $T$ is also [[Definition:Perfectly T4 Space|perfectly $T_4$]].
That is, the property of being [[Definition:Perfectly T4 Space|perfectly $T_4$]] is [[Definition:Hereditar... | {{Recall|Perfectly T4 Space|perfectly $T_4$ space}}
{{:Definition:Perfectly T4 Space}}
Let $H \subseteq S$ be a [[Definition:Topological Subspace|subspace]] of $T$, with the [[Definition:Subspace Topology|subspace topology]] $\tau_H$.
Let $F \subseteq H$ be a [[Definition:Closed Set (Topology)|closed set]] in $H$.
T... | Perfectly T4 Property is Hereditary | https://proofwiki.org/wiki/Perfectly_T4_Property_is_Hereditary | https://proofwiki.org/wiki/Perfectly_T4_Property_is_Hereditary | [
"Perfectly T4 Spaces",
"Zero Sets",
"G-Delta Sets"
] | [
"Definition:Perfectly T4 Space",
"Definition:Topological Subspace",
"Definition:Perfectly T4 Space",
"Definition:Perfectly T4 Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Closed Set/Topology",
"Closed Set in Topological Subspace",
"Definition:Closed Set/Topology",
"Definition:Perfectly T4 Space",
"Characterization of Perfectly T4 Space",
"Definition:Continuous Mapping (Topology)",
"Defi... |
proofwiki-22703 | Perfectly T4 Space is T5 | Every perfectly $T_4$ space is a $T_5$ space. | Let $T = \struct {S, \tau}$ be a perfectly $T_4$ space.
By Perfectly $T_4$ Property is Hereditary, every subspace of $T$ is also perfectly $T_4$.
In particular, every subspace of $T$ is a $T_4$ space.
{{Recall|T5 Space|$T_5$ space|index = 3}}
{{:Definition:T5 Space/Definition 3}}
It follows that $T$ is a $T_5$ space.
{... | Every [[Definition:Perfectly T4 Space|perfectly $T_4$ space]] is a [[Definition:T5 Space|$T_5$ space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]].
By [[Perfectly T4 Property is Hereditary|Perfectly $T_4$ Property is Hereditary]], every [[Definition:Topological Subspace|subspace]] of $T$ is also [[Definition:Perfectly T4 Space|perfectly $T_4$]].
In particular, every [[Defin... | Perfectly T4 Space is T5 | https://proofwiki.org/wiki/Perfectly_T4_Space_is_T5 | https://proofwiki.org/wiki/Perfectly_T4_Space_is_T5 | [
"Perfectly T4 Spaces",
"T5 Spaces"
] | [
"Definition:Perfectly T4 Space",
"Definition:T5 Space"
] | [
"Definition:Perfectly T4 Space",
"Perfectly T4 Property is Hereditary",
"Definition:Topological Subspace",
"Definition:Perfectly T4 Space",
"Definition:Topological Subspace",
"Definition:T4 Space",
"Definition:T5 Space"
] |
proofwiki-22704 | Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Power Series Expansion | The following definition of the concept of the real exponential function:
=== As the Limit of a Sequence ===
{{:Definition:Exponential Function/Real/Limit of Sequence}}
implies the following definition:
=== As a Power Series Expansion ===
{{:Definition:Exponential Function/Real/Power Series Expansion}} | Fix $x \in \R$, and consider the limit:
:$\ds \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$
which is the definition of $\exp x$ as the limit of a sequence.
Since:
:$n \cdot \dfrac x n \to x$
as $n \to \infty$, we have from Generalized Exponential Limit that:
:$\ds \lim_{n \mathop \to \infty} \paren {1 + \frac x... | The following definition of the concept of the [[Definition:Real Exponential Function|real exponential function]]:
=== [[Definition:Exponential Function/Real/Limit of Sequence|As the Limit of a Sequence]] ===
{{:Definition:Exponential Function/Real/Limit of Sequence}}
implies the following definition:
=== [[Definiti... | Fix $x \in \R$, and consider the [[Definition:Limit of Real Sequence|limit]]:
:$\ds \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$
which is the definition of $\exp x$ as the [[Definition:Exponential Function/Real/Limit of Sequence|limit of a sequence]].
Since:
:$n \cdot \dfrac x n \to x$
as $n \to \infty$, we ... | Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Power Series Expansion/Proof 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Exponential_Function/Limit_of_Sequence_implies_Power_Series_Expansion | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Exponential_Function/Limit_of_Sequence_implies_Power_Series_Expansion/Proof_1 | [
"Equivalence of Definitions of Exponential Function"
] | [
"Definition:Exponential Function/Real",
"Definition:Exponential Function/Real/Limit of Sequence",
"Definition:Exponential Function/Real/Power Series Expansion"
] | [
"Definition:Limit of Sequence/Real Numbers",
"Definition:Exponential Function/Real/Limit of Sequence",
"Generalized Exponential Limit",
"Definition:Exponential Function/Real/Power Series Expansion",
"Definition:Exponential Function/Real/Limit of Sequence",
"Definition:Exponential Function/Real/Power Serie... |
proofwiki-22705 | Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Power Series Expansion | The following definition of the concept of the real exponential function:
=== As the Limit of a Sequence ===
{{:Definition:Exponential Function/Real/Limit of Sequence}}
implies the following definition:
=== As a Power Series Expansion ===
{{:Definition:Exponential Function/Real/Power Series Expansion}} | Let $\exp x$ be the real function defined as the limit of the sequence:
:$\exp x := \ds \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$
From the General Binomial Theorem:
{{begin-eqn}}
{{eqn | l = \paren {1 + \frac x n}^n
| r = 1 + x + \frac {n \paren {n - 1} x^2} {2! \ n^2} + \frac {n \paren {n - 1} \paren... | The following definition of the concept of the [[Definition:Real Exponential Function|real exponential function]]:
=== [[Definition:Exponential Function/Real/Limit of Sequence|As the Limit of a Sequence]] ===
{{:Definition:Exponential Function/Real/Limit of Sequence}}
implies the following definition:
=== [[Definiti... | Let $\exp x$ be the [[Definition:Real Function|real function]] defined as the [[Definition:Exponential Function/Real/Limit of Sequence|limit of the sequence]]:
:$\exp x := \ds \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$
From the [[General Binomial Theorem]]:
{{begin-eqn}}
{{eqn | l = \paren {1 + \frac x n}... | Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Power Series Expansion/Proof 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Exponential_Function/Limit_of_Sequence_implies_Power_Series_Expansion | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Exponential_Function/Limit_of_Sequence_implies_Power_Series_Expansion/Proof_2 | [
"Equivalence of Definitions of Exponential Function"
] | [
"Definition:Exponential Function/Real",
"Definition:Exponential Function/Real/Limit of Sequence",
"Definition:Exponential Function/Real/Power Series Expansion"
] | [
"Definition:Real Function",
"Definition:Exponential Function/Real/Limit of Sequence",
"Binomial Theorem/General Binomial Theorem",
"Definition:Basic Null Sequence",
"Power over Factorial"
] |
proofwiki-22706 | Basis Constant is Greater Than or Equal to One | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\GF$ with $X \ne \set { {\mathbf 0}_X}$.
Let $E = \sequence {e_n}_{n \mathop \in \N}$ be a Schauder basis for $X$.
Let $K_b$ be the basis constant for $E$.
Then $K_b \ge 1$. | Let $x \in X$ be such that $\norm x = 1$.
Let $\sequence {S_n}_{n \mathop \in \N}$ be the projections associated with $\sequence {e_n}_{n \mathop \in \N}$.
From Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology and Characterization of Convergence in Strong Operator Topology, we... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]] over $\GF$ with $X \ne \set { {\mathbf 0}_X}$.
Let $E = \sequence {e_n}_{n \mathop \in \N}$ be a [[Definition:Schauder Basis|Schauder basis]] for $X$.
Let $K_b$ be the [[Definition:Basis Constant|basis ... | Let $x \in X$ be such that $\norm x = 1$.
Let $\sequence {S_n}_{n \mathop \in \N}$ be the [[Definition:Projections Associated with Schauder Basis|projections associated with $\sequence {e_n}_{n \mathop \in \N}$]].
From [[Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology]] and... | Basis Constant is Greater Than or Equal to One | https://proofwiki.org/wiki/Basis_Constant_is_Greater_Than_or_Equal_to_One | https://proofwiki.org/wiki/Basis_Constant_is_Greater_Than_or_Equal_to_One | [
"Schauder Bases"
] | [
"Definition:Banach Space",
"Definition:Schauder Basis",
"Definition:Basis Constant"
] | [
"Definition:Projections Associated with Schauder Basis",
"Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology",
"Characterization of Convergence in Strong Operator Topology",
"Modulus of Limit/Normed Vector Space",
"Definition:Space of Bounded Linear Transformations",... |
proofwiki-22707 | Finite Union of Pointwise Equicontinuous Families of Functions is Pointwise Equicontinuous | Let $\struct {X, d_X}$ and $\struct {Y, d_Y}$ be metric spaces.
Let $n \in \N$.
Let $\FF_1, \ldots, \FF_n$ be pointwise equicontinuous families of functions $X \to Y$.
Let:
:$\ds \FF = \bigcup_{j \mathop = 1}^n \FF_j$
Then $\FF$ is pointwise equicontinuous. | Let $x_0 \in X$ and $\epsilon > 0$.
For each $i \in \set {1, \ldots, n}$, there exists $\delta_i > 0$ such that:
:whenever $\map {d_X} {x, x_0} < \delta_i$, we have $\map {d_Y} {\map f x, \map f {x_0} } < \epsilon$ for all $f \in \FF_i$.
Let:
:$\delta = \min \set {\delta_1, \delta_2, \ldots, \delta_n}$
Then for all $x... | Let $\struct {X, d_X}$ and $\struct {Y, d_Y}$ be [[Definition:Metric Space|metric spaces]].
Let $n \in \N$.
Let $\FF_1, \ldots, \FF_n$ be [[Definition:Pointwise Equicontinuous|pointwise equicontinuous families]] of [[Definition:Function|functions]] $X \to Y$.
Let:
:$\ds \FF = \bigcup_{j \mathop = 1}^n \FF_j$
The... | Let $x_0 \in X$ and $\epsilon > 0$.
For each $i \in \set {1, \ldots, n}$, there exists $\delta_i > 0$ such that:
:whenever $\map {d_X} {x, x_0} < \delta_i$, we have $\map {d_Y} {\map f x, \map f {x_0} } < \epsilon$ for all $f \in \FF_i$.
Let:
:$\delta = \min \set {\delta_1, \delta_2, \ldots, \delta_n}$
Then for all... | Finite Union of Pointwise Equicontinuous Families of Functions is Pointwise Equicontinuous | https://proofwiki.org/wiki/Finite_Union_of_Pointwise_Equicontinuous_Families_of_Functions_is_Pointwise_Equicontinuous | https://proofwiki.org/wiki/Finite_Union_of_Pointwise_Equicontinuous_Families_of_Functions_is_Pointwise_Equicontinuous | [
"Pointwise Equicontinuity"
] | [
"Definition:Metric Space",
"Definition:Pointwise Equicontinuous",
"Definition:Function",
"Definition:Pointwise Equicontinuous"
] | [
"Category:Pointwise Equicontinuity"
] |
proofwiki-22708 | Set of Points in Metric Space for which Pointwise Equicontinuous Sequence of Continuous Functions Converges to Continuous Function is Closed | Let $\struct {X, d_X}$ and $\struct {Y, d_Y}$ be metric spaces.
Let $f : X \to Y$ be a continuous function.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a pointwise equicontinuous sequence of continuous functions $X \to Y$.
Let:
:$C = \set {x \in X : \map {f_n} x \to \map f x \text { as } n \to \infty}$
Then $C$ is clo... | From Finite Set of Continuous Functions between Metric Spaces is Pointwise Equicontinuous, $\set f$ is pointwise equicontinuous.
From Finite Union of Pointwise Equicontinuous Families of Functions is Pointwise Equicontinuous, $\set f \cup \set {f_n : n \in \N}$ is pointwise equicontinuous.
From Subset of Metric Space c... | Let $\struct {X, d_X}$ and $\struct {Y, d_Y}$ be [[Definition:Metric Space|metric spaces]].
Let $f : X \to Y$ be a [[Definition:Continuous Function|continuous function]].
Let $\sequence {f_n}_{n \mathop \in \N}$ be a [[Definition:Pointwise Equicontinuous|pointwise equicontinuous]] [[Definition:Sequence|sequence]] of... | From [[Finite Set of Continuous Functions between Metric Spaces is Pointwise Equicontinuous]], $\set f$ is [[Definition:Pointwise Equicontinuous|pointwise equicontinuous]].
From [[Finite Union of Pointwise Equicontinuous Families of Functions is Pointwise Equicontinuous]], $\set f \cup \set {f_n : n \in \N}$ is [[Defi... | Set of Points in Metric Space for which Pointwise Equicontinuous Sequence of Continuous Functions Converges to Continuous Function is Closed | https://proofwiki.org/wiki/Set_of_Points_in_Metric_Space_for_which_Pointwise_Equicontinuous_Sequence_of_Continuous_Functions_Converges_to_Continuous_Function_is_Closed | https://proofwiki.org/wiki/Set_of_Points_in_Metric_Space_for_which_Pointwise_Equicontinuous_Sequence_of_Continuous_Functions_Converges_to_Continuous_Function_is_Closed | [
"Pointwise Equicontinuity"
] | [
"Definition:Metric Space",
"Definition:Continuous Function",
"Definition:Pointwise Equicontinuous",
"Definition:Sequence",
"Definition:Continuous Function",
"Definition:Closed Set"
] | [
"Finite Set of Continuous Functions between Metric Spaces is Pointwise Equicontinuous",
"Definition:Pointwise Equicontinuous",
"Finite Union of Pointwise Equicontinuous Families of Functions is Pointwise Equicontinuous",
"Definition:Pointwise Equicontinuous",
"Subset of Metric Space contains Limits of Seque... |
proofwiki-22709 | Uniformly Bounded Sequence of Bounded Linear Transformations is Uniformly Equicontinuous | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces.
Let $\map \BB {X, Y}$ be the space of bounded linear transformations $X \to Y$.
Let $\norm {\, \cdot \,}_{\map \BB {X, Y} }$ be the norm on $\map \BB {X, Y}$.
Let $\sequence {T_n}_{n \... | Let $x, y \in X$.
Let:
:$\ds M = \sup_{n \mathop \in \N} \norm {T_n}_{\map \BB {X, Y} }$
Fix $n \in \N$.
From Fundamental Property of Norm on Bounded Linear Transformation, we have:
:$\ds \norm {T_n x - T_n y}_Y = \norm {\map {T_n} {x - y} }_Y \le \norm {T_n}_{\map \BB {X, Y} } \norm {x - y}_X$
We have:
:$\norm {T_n}_{... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $\map \BB {X, Y}$ be the [[Definition:Space of Bounded Linear Transformations|space of bounded linear transformations $X \to Y$]].
Let $\norm ... | Let $x, y \in X$.
Let:
:$\ds M = \sup_{n \mathop \in \N} \norm {T_n}_{\map \BB {X, Y} }$
Fix $n \in \N$.
From [[Fundamental Property of Norm on Bounded Linear Transformation]], we have:
:$\ds \norm {T_n x - T_n y}_Y = \norm {\map {T_n} {x - y} }_Y \le \norm {T_n}_{\map \BB {X, Y} } \norm {x - y}_X$
We have:
:$\norm... | Uniformly Bounded Sequence of Bounded Linear Transformations is Uniformly Equicontinuous | https://proofwiki.org/wiki/Uniformly_Bounded_Sequence_of_Bounded_Linear_Transformations_is_Uniformly_Equicontinuous | https://proofwiki.org/wiki/Uniformly_Bounded_Sequence_of_Bounded_Linear_Transformations_is_Uniformly_Equicontinuous | [
"Uniform Equicontinuity"
] | [
"Definition:Normed Vector Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Sequence",
"Definition:Uniformly Equicontinuous"
] | [
"Fundamental Property of Norm on Bounded Linear Transformation",
"Definition:Lipschitz Continuity",
"Family of Lipschitz Continuous Functions with same Lipschitz Constant is Uniformly Equicontinuous",
"Definition:Uniformly Equicontinuous",
"Category:Uniform Equicontinuity"
] |
proofwiki-22710 | Construction of Schauder Basis from Bounded Projections | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be an infinite-dimensional normed vector space over $\GF$.
Let $\sequence {S_n}_{n \mathop \in \N}$ be a sequence of bounded linear transformations such that:
:$(1) \quad$ $\dim S_n \sqbrk X = n$ for all $n \in \N$
:$(2) \quad$ $S_n S_m = S_m S_n = S_... | Let $\BB X$ be the space of bounded linear transformations on $X$.
Let $\norm {\, \cdot \,}_{\map \BB X}$ be the norm on $\map \BB X$.
From Scalar Multiple of Schauder Basis is Schauder Basis, $\sequence {e_n}_{n \mathop \in \N}$ is a Schauder basis {{iff}} $\ds \sequence {\frac {e_n} {\norm {e_n} } }_{n \mathop \in \... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\sequence {S_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Bounded Lin... | Let $\BB X$ be the [[Definition:Space of Bounded Linear Transformations|space of bounded linear transformations on $X$]].
Let $\norm {\, \cdot \,}_{\map \BB X}$ be the [[Definition:Norm on Bounded Linear Transformation|norm on $\map \BB X$]].
From [[Scalar Multiple of Schauder Basis is Schauder Basis]], $\sequence {... | Construction of Schauder Basis from Bounded Projections | https://proofwiki.org/wiki/Construction_of_Schauder_Basis_from_Bounded_Projections | https://proofwiki.org/wiki/Construction_of_Schauder_Basis_from_Bounded_Projections | [
"Schauder Bases"
] | [
"Definition:Infinite-Dimensional Vector Space",
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Bounded Linear Transformation",
"Definition:Schauder Basis",
"Definition:Projections Associated with Schauder Basis"
] | [
"Definition:Space of Bounded Linear Transformations",
"Definition:Norm/Bounded Linear Transformation",
"Scalar Multiple of Schauder Basis is Schauder Basis",
"Definition:Schauder Basis",
"Definition:Schauder Basis",
"Condition for Sequence to be Schauder Basis in terms of Coordinate Functionals",
"Defin... |
proofwiki-22711 | Condition for Sequence to be Schauder Basis in terms of Coordinate Functionals | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be an infinite-dimensional normed vector space over $\GF$.
Let $\sequence {e_n}_{n \mathop \in \N}$ be a sequence in $X$ such that there exists bounded linear functionals $\sequence {e_n^\ast}_{n \mathop \in \N}$ on $X$ such that:
:$(1) \quad$ $\map ... | It remains to verify that for each $x \in X$, if:
:$\ds \sum_{n \mathop = 1}^\infty \alpha_n e_n = x$
for some sequence $\sequence {\alpha_n}_{n \mathop \in \N}$ in $\GF$, then:
:$\alpha_n = \map {e_n^\ast} x$
Then we will have that:
:$\ds \sum_{n \mathop = 1}^\infty \alpha_n e_n = x$
is satisfied for a unique sequence... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\sequence {e_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$ such that there exi... | It remains to verify that for each $x \in X$, if:
:$\ds \sum_{n \mathop = 1}^\infty \alpha_n e_n = x$
for some [[Definition:Sequence|sequence]] $\sequence {\alpha_n}_{n \mathop \in \N}$ in $\GF$, then:
:$\alpha_n = \map {e_n^\ast} x$
Then we will have that:
:$\ds \sum_{n \mathop = 1}^\infty \alpha_n e_n = x$
is satisf... | Condition for Sequence to be Schauder Basis in terms of Coordinate Functionals | https://proofwiki.org/wiki/Condition_for_Sequence_to_be_Schauder_Basis_in_terms_of_Coordinate_Functionals | https://proofwiki.org/wiki/Condition_for_Sequence_to_be_Schauder_Basis_in_terms_of_Coordinate_Functionals | [
"Schauder Bases"
] | [
"Definition:Infinite-Dimensional Vector Space",
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Bounded Linear Functional",
"Definition:Kronecker Delta/Number",
"Definition:Schauder Basis"
] | [
"Definition:Sequence",
"Definition:Sequence",
"Definition:Schauder Basis",
"Definition:Bounded Linear Functional",
"Sequential Continuity is Equivalent to Continuity in Metric Space",
"Category:Schauder Bases"
] |
proofwiki-22712 | Grunblum's Criterion | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\GF$.
Let $\sequence {e_j}_{j \mathop \in \N}$ be a sequence of non-zero vectors in $X$.
Then $\sequence {e_j}_{j \mathop \in \N}$ is a basic sequence {{iff}} there exists $K > 0$ such that for any sequence $\sequence {\alpha_j... | Let $\map \BB X$ be the space of bounded linear transformations.
Let $\norm {\, \cdot \,}_{\map \BB X}$ be the norm on $\map \BB X$. | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\sequence {e_j}_{j \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Zero Vector|non-zero vectors]] in $X$.
Then $\sequence {e_j}_{j \mathop \in \N}$ is a [[Defini... | Let $\map \BB X$ be the [[Definition:Space of Bounded Linear Transformations|space of bounded linear transformations]].
Let $\norm {\, \cdot \,}_{\map \BB X}$ be the [[Definition:Norm on Bounded Linear Transformation|norm]] on $\map \BB X$. | Grunblum's Criterion | https://proofwiki.org/wiki/Grunblum's_Criterion | https://proofwiki.org/wiki/Grunblum's_Criterion | [
"Basic Sequences"
] | [
"Definition:Banach Space",
"Definition:Sequence",
"Definition:Zero Vector",
"Definition:Basic Sequence",
"Definition:Sequence",
"Definition:Basic Sequence",
"Definition:Basis Constant"
] | [
"Definition:Space of Bounded Linear Transformations",
"Definition:Norm/Bounded Linear Transformation"
] |
proofwiki-22713 | Basic Sequences in Banach Spaces are Equivalent iff Map between is Linear Isomorphism | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be Banach spaces over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a basic sequence in $\struct {X, \norm {\, \cdot \,}_X}$ and $\sequence {y_n}_{n \mathop \in \N}$ be a basic sequence in $\struct {Y, \n... | {{WLOG}} suppose that $X = \sqbrk {x_n}_{n \mathop \in \N}$ and $Y = \sqbrk {y_n}_{n \mathop \in \N}$.
We are assured that $X$ and $Y$ are still Banach spaces by Closed Subspace of Banach Space forms Banach Space. | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Banach Space|Banach spaces]] over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Basic Sequence|basic sequence]] in $\struct {X, \norm {\, \cdot \,}_X}$ and $\sequence {y_n... | {{WLOG}} suppose that $X = \sqbrk {x_n}_{n \mathop \in \N}$ and $Y = \sqbrk {y_n}_{n \mathop \in \N}$.
We are assured that $X$ and $Y$ are still [[Definition:Banach Space|Banach spaces]] by [[Closed Subspace of Banach Space forms Banach Space]]. | Basic Sequences in Banach Spaces are Equivalent iff Map between is Linear Isomorphism | https://proofwiki.org/wiki/Basic_Sequences_in_Banach_Spaces_are_Equivalent_iff_Map_between_is_Linear_Isomorphism | https://proofwiki.org/wiki/Basic_Sequences_in_Banach_Spaces_are_Equivalent_iff_Map_between_is_Linear_Isomorphism | [
"Basic Sequences"
] | [
"Definition:Banach Space",
"Definition:Basic Sequence",
"Definition:Basic Sequence",
"Definition:Closed Linear Span",
"Definition:Equivalence of Basic Sequences",
"Definition:Linear Isomorphism"
] | [
"Definition:Banach Space",
"Closed Subspace of Banach Space forms Banach Space"
] |
proofwiki-22714 | Block Basic Sequence is Basic Sequence | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\GF$.
Let $\sequence {e_n}_{n \mathop \in \N}$ be a Schauder basis for $\struct {X, \norm {\, \cdot \,} }$.
Let $K_b$ be the basis constant of $\sequence {e_n}_{n \mathop \in \N}$.
Let $\sequence {p_n}_{n \mathop \in \N}$ be a ... | Let $\sequence {\beta_k}_{k \mathop \in \N}$ be a sequence in $\GF$.
Then we have:
:$\ds \sum_{k \mathop = 1}^m \beta_k u_k = \sum_{k \mathop = 1}^m \beta_k \sum_{j \mathop = p_{k - 1} + 1}^{p_k} \alpha_j e_j = \sum_{k \mathop = 1}^m \sum_{j \mathop = p_{k - 1} + 1}^{p_k} \beta_k \alpha_j e_j$
Taking $\gamma_j = \alpha... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\sequence {e_n}_{n \mathop \in \N}$ be a [[Definition:Schauder Basis|Schauder basis]] for $\struct {X, \norm {\, \cdot \,} }$.
Let $K_b$ be the [[Definition:Basis Constant|basis constan... | Let $\sequence {\beta_k}_{k \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\GF$.
Then we have:
:$\ds \sum_{k \mathop = 1}^m \beta_k u_k = \sum_{k \mathop = 1}^m \beta_k \sum_{j \mathop = p_{k - 1} + 1}^{p_k} \alpha_j e_j = \sum_{k \mathop = 1}^m \sum_{j \mathop = p_{k - 1} + 1}^{p_k} \beta_k \alpha_j e_j$
... | Block Basic Sequence is Basic Sequence | https://proofwiki.org/wiki/Block_Basic_Sequence_is_Basic_Sequence | https://proofwiki.org/wiki/Block_Basic_Sequence_is_Basic_Sequence | [
"Block Basic Sequences",
"Basic Sequences",
"Block Basic Sequences"
] | [
"Definition:Banach Space",
"Definition:Schauder Basis",
"Definition:Basis Constant",
"Definition:Strictly Increasing/Sequence",
"Definition:Sequence",
"Definition:Block Basic Sequence",
"Definition:Basic Sequence",
"Definition:Basis Constant"
] | [
"Definition:Sequence",
"Grunblum's Criterion",
"Definition:Sequence",
"Grunblum's Criterion",
"Definition:Basic Sequence"
] |
proofwiki-22715 | Principle of Small Perturbations | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a basic sequence with basis constant $K_b$.
Let $\sequence {y_n}_{n \mathop \in \N}$ be a sequence in $X$ with:
:$\ds 2 K_b \sum_{n \mathop = 1}^\infty \frac {\norm {x_n - y_n} ... | Let $Z = \sqbrk {x_n}_{n \mathop \in \N}$ be the closed linear span of $\sequence {x_n}_{n \mathop \in \N}$.
Let $\struct {Z^\ast, \norm {\, \cdot \,}_{Z^\ast} }$ be the normed dual space of $Z$.
Let $\map \BB Z$ be the space of bounded linear transformations on $Z$.
Let $\norm {\, \cdot \,}_{\map \BB Z}$ be the norm o... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Basic Sequence|basic sequence]] with [[Definition:Basis Constant|basis constant]] $K_b$.
Let $\sequence {y_n}_{n \mathop \in \N}$ b... | Let $Z = \sqbrk {x_n}_{n \mathop \in \N}$ be the [[Definition:Closed Linear Span|closed linear span]] of $\sequence {x_n}_{n \mathop \in \N}$.
Let $\struct {Z^\ast, \norm {\, \cdot \,}_{Z^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $Z$.
Let $\map \BB Z$ be the [[Definition:Space of Bounded L... | Principle of Small Perturbations | https://proofwiki.org/wiki/Principle_of_Small_Perturbations | https://proofwiki.org/wiki/Principle_of_Small_Perturbations | [
"Basic Sequences"
] | [
"Definition:Banach Space",
"Definition:Basic Sequence",
"Definition:Basis Constant",
"Definition:Sequence",
"Definition:Congruent Sequences",
"Definition:Basic Sequence",
"Definition:Basis Constant"
] | [
"Definition:Closed Linear Span",
"Definition:Normed Dual Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Coordinate Functionals Associated with Schauder Basis",
"Definition:Projections Associated with Schauder Basis",
"Fundamenta... |
proofwiki-22716 | Bessaga-Pełczyński Selection Principle | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\GF$.
Let $\sequence {e_n}_{n \mathop \in \N}$ be a Schauder basis for $\struct {X, \norm {\, \cdot \,} }$ with basis constant $K_b$.
Let $\sequence {e_n^\ast}_{n \mathop \in \N}$ be the coordinate functionals associated with ... | Let:
:$\ds \alpha = \inf_n \norm {x_n} > 0$
Let $\nu \in \openint 0 {1/4}$.
Take $n_1 = 1$ and $r_0 = 0$.
Let $\sequence {S_n}_{n \mathop \in \N}$ be the projections associated with $\sequence {e_n}_{n \mathop \in \N}$.
From Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology: Co... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\sequence {e_n}_{n \mathop \in \N}$ be a [[Definition:Schauder Basis|Schauder basis]] for $\struct {X, \norm {\, \cdot \,} }$ with [[Definition:Basis Constant|basis constant]] $K_b$.
L... | Let:
:$\ds \alpha = \inf_n \norm {x_n} > 0$
Let $\nu \in \openint 0 {1/4}$.
Take $n_1 = 1$ and $r_0 = 0$.
Let $\sequence {S_n}_{n \mathop \in \N}$ be the [[Definition:Projections Associated with Schauder Basis|projections associated with $\sequence {e_n}_{n \mathop \in \N}$]].
From [[Projections Associated with Sch... | Bessaga-Pełczyński Selection Principle | https://proofwiki.org/wiki/Bessaga-Pełczyński_Selection_Principle | https://proofwiki.org/wiki/Bessaga-Pełczyński_Selection_Principle | [
"Basic Sequences"
] | [
"Definition:Banach Space",
"Definition:Schauder Basis",
"Definition:Basis Constant",
"Definition:Coordinate Functionals Associated with Schauder Basis",
"Definition:Sequence",
"Definition:Subsequence",
"Definition:Congruent Sequences",
"Definition:Block Basic Sequence",
"Definition:Subsequence",
"... | [
"Definition:Projections Associated with Schauder Basis",
"Projections Associated with Schauder Basis Converge to Identity in Strong Operator Topology/Corollary",
"Linear Combination of Convergent Sequences in Topological Vector Space is Convergent",
"Multiple of Vector in Topological Vector Space Converges",
... |
proofwiki-22717 | Orthonormal Basis for Separable Hilbert Space is Monotone Basis | Let $\GF \in \set {\R, \C}$.
Let $\HH$ be a separable Hilbert space over $\GF$.
Let $\sequence {e_n}_{n \mathop \in \N}$ be an orthonormal basis for $\HH$.
Then $\sequence {e_n}_{n \mathop \in \N}$ is a monotone basis. | Let $\map \BB \HH$ be the space of bounded linear transformations on $\HH$.
Let $\norm {\, \cdot \,}_{\map \BB \HH}$ be the norm on $\map \BB \HH$.
We firstly show that $\sequence {e_n}_{n \mathop \in \N}$ is a Schauder basis.
For each $n \in \N$, define $e_n^\ast : \HH \to \GF$ by:
:$\map {e_n^\ast} x = \innerprod x ... | Let $\GF \in \set {\R, \C}$.
Let $\HH$ be a [[Definition:Separable Space|separable]] [[Definition:Hilbert Space|Hilbert space]] over $\GF$.
Let $\sequence {e_n}_{n \mathop \in \N}$ be an [[Definition:Orthonormal Basis|orthonormal basis]] for $\HH$.
Then $\sequence {e_n}_{n \mathop \in \N}$ is a [[Definition:Monot... | Let $\map \BB \HH$ be the [[Definition:Space of Bounded Linear Transformations|space of bounded linear transformations on $\HH$]].
Let $\norm {\, \cdot \,}_{\map \BB \HH}$ be the [[Definition:Norm on Bounded Linear Transformation|norm on $\map \BB \HH$]].
We firstly show that $\sequence {e_n}_{n \mathop \in \N}$ is ... | Orthonormal Basis for Separable Hilbert Space is Monotone Basis | https://proofwiki.org/wiki/Orthonormal_Basis_for_Separable_Hilbert_Space_is_Monotone_Basis | https://proofwiki.org/wiki/Orthonormal_Basis_for_Separable_Hilbert_Space_is_Monotone_Basis | [
"Monotone Bases",
"Hilbert Spaces"
] | [
"Definition:Separable Space",
"Definition:Hilbert Space",
"Definition:Orthonormal Basis",
"Definition:Monotone Basis"
] | [
"Definition:Space of Bounded Linear Transformations",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Schauder Basis",
"Definition:Orthogonal (Linear Algebra)/Sets",
"Definition:Kronecker Delta",
"Riesz Representation Theorem (Hilbert Spaces)",
"Definition:Bounded Linear Functional",
"Cha... |
proofwiki-22718 | Subset of Banach Space Bounded Away from Zero including Zero Vector in Weak Closure contains Basic Sequence with Basis Constant Arbitrarily Close to 1 | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $w$ be the weak topology on $X$.
Let $\cl_w$ be the closure taken in $\struct {X, w}$.
Let $S \subseteq X$ be such that:
:$\ds \inf_{x \mathop \in S} \norm x_X > 0$
and:
:${\mathbf 0}_X \in \map {\cl_w} S$
Let $\eps... | For convenience, we first prove a lemma in the normed dual space of $X$. | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology on $X$]].
Let $\cl_w$ be the [[Definition:Topological Closure|closure]] taken in $\struct {X, w}$.
Let ... | For convenience, we first prove a lemma in the [[Definition:Normed Dual Space|normed dual space]] of $X$. | Subset of Banach Space Bounded Away from Zero including Zero Vector in Weak Closure contains Basic Sequence with Basis Constant Arbitrarily Close to 1 | https://proofwiki.org/wiki/Subset_of_Banach_Space_Bounded_Away_from_Zero_including_Zero_Vector_in_Weak_Closure_contains_Basic_Sequence_with_Basis_Constant_Arbitrarily_Close_to_1 | https://proofwiki.org/wiki/Subset_of_Banach_Space_Bounded_Away_from_Zero_including_Zero_Vector_in_Weak_Closure_contains_Basic_Sequence_with_Basis_Constant_Arbitrarily_Close_to_1 | [
"Weak-* Topologies",
"Basic Sequences",
"Subset of Banach Space Bounded Away from Zero including Zero Vector in Weak Closure contains Basic Sequence with Basis Constant Arbitrarily Close to 1"
] | [
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Closure (Topology)",
"Definition:Basic Sequence",
"Definition:Basis Constant"
] | [
"Definition:Normed Dual Space"
] |
proofwiki-22719 | Equations of Circular Motion in Cartesian Plane/Polar Form | Let:
{{begin-eqn}}
{{eqn | n = 1
| l = \mathbf e_r
| r = \mathbf i \cos \theta + \mathbf j \sin \theta
}}
{{eqn | n = 2
| l = \mathbf e_\theta
| r = -\mathbf i \sin \theta + \mathbf j \cos \theta
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \mathbf r
| r = r_0 \mathbf e_r
}}
{{eqn | l = ... | From Equations of Circular Motion in Cartesian Plane:
{{begin-eqn}}
{{eqn | l = \mathbf r
| r = r_0 \paren {\mathbf i \cos \theta + \mathbf j \sin \theta}
}}
{{eqn | l = \mathbf v = \dot {\mathbf r}
| r = r_0 \paren {-\dot \theta \mathbf i \sin \theta + \dot \theta \mathbf j \cos \theta}
}}
{{eqn | l = \mat... | Let:
{{begin-eqn}}
{{eqn | n = 1
| l = \mathbf e_r
| r = \mathbf i \cos \theta + \mathbf j \sin \theta
}}
{{eqn | n = 2
| l = \mathbf e_\theta
| r = -\mathbf i \sin \theta + \mathbf j \cos \theta
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \mathbf r
| r = r_0 \mathbf e_r
}}
{{eqn | l... | From [[Equations of Circular Motion in Cartesian Plane]]:
{{begin-eqn}}
{{eqn | l = \mathbf r
| r = r_0 \paren {\mathbf i \cos \theta + \mathbf j \sin \theta}
}}
{{eqn | l = \mathbf v = \dot {\mathbf r}
| r = r_0 \paren {-\dot \theta \mathbf i \sin \theta + \dot \theta \mathbf j \cos \theta}
}}
{{eqn | l =... | Equations of Circular Motion in Cartesian Plane/Polar Form | https://proofwiki.org/wiki/Equations_of_Circular_Motion_in_Cartesian_Plane/Polar_Form | https://proofwiki.org/wiki/Equations_of_Circular_Motion_in_Cartesian_Plane/Polar_Form | [
"Circular Motion"
] | [] | [
"Equations of Circular Motion in Cartesian Plane"
] |
proofwiki-22720 | Weak-* Closure of Isometric Copy of Set in Second Dual contains Isometric Copy of Weak Closure of Set in Original Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $S \subseteq X$.
Let $w$ be the weak topology on $X$.
Let $\cl_w$ be the closure taken in $\struct {X, w}$.
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the second normed dual of $\struct... | From Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Homeomorphism onto Image, $\iota : \struct {X, w} \to \struct {\iota X, w^\ast}$ is a homeomorphism.
From Homeomorphism iff Image of Closure equals Closure of Image and Closure of Subset in Subspace, we have:
:$\map \iota {\map {\cl_w} S} = ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $S \subseteq X$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology on $X$]].
Let $\cl_w$ be the [[Definition:Topological Closure|closure]] taken in $... | From [[Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Homeomorphism onto Image]], $\iota : \struct {X, w} \to \struct {\iota X, w^\ast}$ is a [[Definition:Homeomorphism|homeomorphism]].
From [[Homeomorphism iff Image of Closure equals Closure of Image]] and [[Closure of Subset in Subspace]],... | Weak-* Closure of Isometric Copy of Set in Second Dual contains Isometric Copy of Weak Closure of Set in Original Space | https://proofwiki.org/wiki/Weak-*_Closure_of_Isometric_Copy_of_Set_in_Second_Dual_contains_Isometric_Copy_of_Weak_Closure_of_Set_in_Original_Space | https://proofwiki.org/wiki/Weak-*_Closure_of_Isometric_Copy_of_Set_in_Second_Dual_contains_Isometric_Copy_of_Weak_Closure_of_Set_in_Original_Space | [
"Weak-* Topologies",
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Closure (Topology)",
"Definition:Second Normed Dual",
"Definition:Weak-* Topology",
"Definition:Closure (Topology)",
"Definition:Evaluation Linear Transformation"
] | [
"Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Homeomorphism onto Image",
"Definition:Homeomorphism",
"Homeomorphism iff Image of Closure equals Closure of Image",
"Closure of Subset in Subspace",
"Category:Weak-* Topologies",
"Category:Weak Topologies on Topological Vector Spa... |
proofwiki-22721 | Dimension of Quotient Vector Space | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $Y$ be a vector subspace of $X$.
Let $X/Y$ be the quotient vector space of $X$ modulo $Y$.
Then we have:
:$\dim Y + \map \dim {X/Y} = \dim X$
Hence $X$ is finite-dimensional {{iff}} both $Y$ and $X/Y$ are. | We show that there exists disjoint sets $\BB_1, \BB_2$ such that:
:$\size {\BB_1} = \dim Y$
and:
:$\size {\BB_2} = \map \dim {X/Y}$
such that $\BB_1 \cup \BB_2$ is a basis for $X$.
By the definition of dimension and the cardinal sum, we will then have:
:$\dim Y + \map \dim {X/Y} = \dim X$
Let $q : X \to X/Y$ be the quo... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $Y$ be a [[Definition:Vector Subspace|vector subspace]] of $X$.
Let $X/Y$ be the [[Definition:Quotient Vector Space|quotient vector space of $X$ modulo $Y$]].
Then we have:
:$\dim Y + \ma... | We show that there exists [[Definition:Disjoint Sets|disjoint sets]] $\BB_1, \BB_2$ such that:
:$\size {\BB_1} = \dim Y$
and:
:$\size {\BB_2} = \map \dim {X/Y}$
such that $\BB_1 \cup \BB_2$ is a [[Definition:Basis of Vector Space|basis]] for $X$.
By the definition of [[Definition:Dimension of Vector Space|dimension]] ... | Dimension of Quotient Vector Space | https://proofwiki.org/wiki/Dimension_of_Quotient_Vector_Space | https://proofwiki.org/wiki/Dimension_of_Quotient_Vector_Space | [
"Quotient Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Vector Subspace",
"Definition:Quotient Vector Space",
"Definition:Dimension of Vector Space/Finite"
] | [
"Definition:Disjoint Sets",
"Definition:Basis of Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Sum of Cardinals",
"Definition:Quotient Mapping",
"Definition:Basis of Vector Space",
"Definition:Indexing Set",
"Definition:Cardinality",
"Definition:Basis of Vector Space",
"Defini... |
proofwiki-22722 | Weak Closure of Unit Sphere is Closed Unit Ball | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be an infinite-dimensional normed vector space over $\GF$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\cl_w$ be the topological closure in $\struct {X, w}$.
Let:
:$S_X^- = \set {x \in X : \norm x_X = 1}$
Let $B_X^-$ be... | We firstly show that $\map {\cl_w} {S_X^-} \subseteq B_X^-$.
We have $S_X^- \subseteq B_X^-$.
From Mazur's Theorem: Corollary, $B_X^-$ is $w$-closed.
Hence from Set is Closed iff Equals Topological Closure, we have that $\map {\cl_w} {B_X^-} = B_X^-$.
From Topological Closure of Subset is Subset of Topological Closure... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology on $\struct {X, \norm ... | We firstly show that $\map {\cl_w} {S_X^-} \subseteq B_X^-$.
We have $S_X^- \subseteq B_X^-$.
From [[Mazur's Theorem/Corollary|Mazur's Theorem: Corollary]], $B_X^-$ is [[Definition:Weakly Closed Set|$w$-closed]].
Hence from [[Set is Closed iff Equals Topological Closure]], we have that $\map {\cl_w} {B_X^-} = B_X^-... | Weak Closure of Unit Sphere is Closed Unit Ball/Proof 1 | https://proofwiki.org/wiki/Weak_Closure_of_Unit_Sphere_is_Closed_Unit_Ball | https://proofwiki.org/wiki/Weak_Closure_of_Unit_Sphere_is_Closed_Unit_Ball/Proof_1 | [
"Weak Topologies on Topological Vector Spaces",
"Weak Closure of Unit Sphere is Closed Unit Ball"
] | [
"Definition:Infinite-Dimensional Vector Space",
"Definition:Normed Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Closure (Topology)",
"Definition:Closed Unit Ball"
] | [
"Mazur's Theorem/Corollary",
"Definition:Weakly Closed Set",
"Set is Closed iff Equals Topological Closure",
"Topological Closure of Subset is Subset of Topological Closure",
"Definition:Open Neighborhood",
"Definition:Intersection",
"Definition:Open Neighborhood",
"Translation of Open Set in Topologi... |
proofwiki-22723 | Dimension of Direct Product of Vector Spaces | Let $K$ be a field.
Let $X_1, X_2, \ldots, X_n$ be vector spaces over $K$.
Let:
:$\ds X = \prod_{j \mathop = 1}^n X_j$ be the direct product of $X_1, \ldots, X_n$.
Then:
:$\ds \dim X = \sum_{j \mathop = 1}^n \map \dim {X_j}$ | For each $1 \le j \le n$, let $\BB_j = \sequence {e_{j, i} }_{i \mathop \in I_j}$ be a basis of $X_j$ where $I_j$ is an index set.
{{WLOG}} suppose that the $I_j$ are disjoint.
For each $1 \le j \le n$ and $i \in I_j$, define $\pi : \set {1, \ldots, n} \to X$ by:
:$\map {\pi_{j, i} } k = \begin{cases}{\mathbf 0}_{X_j} ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X_1, X_2, \ldots, X_n$ be [[Definition:Vector Space|vector spaces]] over $K$.
Let:
:$\ds X = \prod_{j \mathop = 1}^n X_j$ be the [[Definition:Direct Product of Vector Spaces|direct product of $X_1, \ldots, X_n$]].
Then:
:$\ds \dim X = \sum_{j \mathop ... | For each $1 \le j \le n$, let $\BB_j = \sequence {e_{j, i} }_{i \mathop \in I_j}$ be a [[Definition:Basis of Vector Space|basis]] of $X_j$ where $I_j$ is an [[Definition:Index Set|index set]].
{{WLOG}} suppose that the $I_j$ are [[Definition:Disjoint Sets|disjoint]].
For each $1 \le j \le n$ and $i \in I_j$, define $... | Dimension of Direct Product of Vector Spaces | https://proofwiki.org/wiki/Dimension_of_Direct_Product_of_Vector_Spaces | https://proofwiki.org/wiki/Dimension_of_Direct_Product_of_Vector_Spaces | [
"Direct Product of Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Direct Product of Vector Spaces"
] | [
"Definition:Basis of Vector Space",
"Definition:Indexing Set",
"Definition:Disjoint Sets",
"Definition:Generator (Linear Algebra)",
"Definition:Linearly Independent/Set",
"Definition:Linearly Independent/Set",
"Definition:Linearly Independent/Set",
"Category:Direct Product of Vector Spaces"
] |
proofwiki-22724 | Frame Spectrum Functor is Contravariant Functor | Let $\mathbf{Frm}$ denote the category of frames.
Let $\mathbf{Top}$ denote the category of topological spaces.
Then:
:the frame spectrum functor $\mathbf {\operatorname {Sp}} : \mathbf{Frm} \to \mathbf{Top}$ is a contravariant functor | === Object Functor is Well-Defined ===
From Spectrum of Locale as Completely Prime Filters is Sober Space:
:for each frame $L$ in $\mathbf {Frm}$, considered as a locale, $\map {\mathbf {\operatorname {Sp} } } L$ is a topological space
It follows that the object functor $\mathbf {\operatorname {Sp}}_0$ is well-defined.... | Let $\mathbf{Frm}$ denote the [[Definition:Category of Frames|category of frames]].
Let $\mathbf{Top}$ denote the [[Definition:Category of Topological Spaces|category of topological spaces]].
Then:
:the [[Definition:Frame Spectrum Functor|frame spectrum functor]] $\mathbf {\operatorname {Sp}} : \mathbf{Frm} \to \mat... | === Object Functor is Well-Defined ===
From [[Spectrum of Locale as Completely Prime Filters is Sober Space]]:
:for each [[Definition:Frame (Lattice Theory)|frame]] $L$ in $\mathbf {Frm}$, considered as a [[Definition:Locale (Lattice Theory)|locale]], $\map {\mathbf {\operatorname {Sp} } } L$ is a [[Definition:Topolog... | Frame Spectrum Functor is Contravariant Functor | https://proofwiki.org/wiki/Frame_Spectrum_Functor_is_Contravariant_Functor | https://proofwiki.org/wiki/Frame_Spectrum_Functor_is_Contravariant_Functor | [
"Functors"
] | [
"Definition:Category of Frames",
"Definition:Category of Topological Spaces",
"Definition:Frame Spectrum Functor",
"Definition:Functor/Contravariant"
] | [
"Spectrum of Locale as Completely Prime Filters is Sober Space",
"Definition:Frame (Lattice Theory)",
"Definition:Locale (Lattice Theory)",
"Definition:Topological Space",
"Definition:Object Functor",
"Definition:Well-Defined",
"Definition:Well-Defined",
"Definition:Frame (Lattice Theory)",
"Definit... |
proofwiki-22725 | Construction of Basis for Vector Space from Basis for Vector Subspace and Basis of Quotient | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $Y$ be a vector subspace of $X$.
Let $X/Y$ be the quotient vector space of $X$ modulo $Y$.
Let $q : X \to X/Y$ be the quotient mapping.
Let $\BB_1 = \set {y_\alpha : \alpha \in I_1}$ be a basis for $Y$, where $I_1$ is an index set with cardinality $\dim Y$.
... | For $\beta, \beta' \in I_2$ with $\beta \ne \beta'$, we have $\map q {z_\beta} \ne \map q {z_{\beta'} }$ and hence $z_\beta \ne z_{\beta'}$.
We first establish that $\BB_1$ and $\BB_2$ are disjoint.
From Kernel of Quotient Mapping, we have that $\map q {y_\alpha} = {\mathbf 0}_{X/Y}$ for each $\alpha \in I_1$.
Since $... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $Y$ be a [[Definition:Vector Subspace|vector subspace]] of $X$.
Let $X/Y$ be the [[Definition:Quotient Vector Space|quotient vector space of $X$ modulo $Y$]].
Let $q : X \to X/Y$ be the [[... | For $\beta, \beta' \in I_2$ with $\beta \ne \beta'$, we have $\map q {z_\beta} \ne \map q {z_{\beta'} }$ and hence $z_\beta \ne z_{\beta'}$.
We first establish that $\BB_1$ and $\BB_2$ are [[Definition:Disjoint Sets|disjoint]].
From [[Kernel of Quotient Mapping]], we have that $\map q {y_\alpha} = {\mathbf 0}_{X/Y}$ ... | Construction of Basis for Vector Space from Basis for Vector Subspace and Basis of Quotient | https://proofwiki.org/wiki/Construction_of_Basis_for_Vector_Space_from_Basis_for_Vector_Subspace_and_Basis_of_Quotient | https://proofwiki.org/wiki/Construction_of_Basis_for_Vector_Space_from_Basis_for_Vector_Subspace_and_Basis_of_Quotient | [
"Quotient Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Vector Subspace",
"Definition:Quotient Vector Space",
"Definition:Quotient Mapping",
"Definition:Basis of Vector Space",
"Definition:Indexing Set",
"Definition:Cardinality",
"Definition:Basis of Vector Space",
"Definitio... | [
"Definition:Disjoint Sets",
"Kernel of Quotient Mapping",
"Definition:Basis of Vector Space",
"Definition:Linearly Independent/Set",
"Definition:Linearly Independent/Set",
"Kernel of Quotient Mapping",
"Quotient Mapping is Linear Transformation",
"Definition:Linearly Independent/Set",
"Definition:Li... |
proofwiki-22726 | Characterization of Codimension in terms of Direct Sum | Let $K$ be a field.
Let $X$ be a vector space such that $X \ne {\mathbf 0}_X$.
Let $U \subseteq X$ be a vector subspace of $X$.
Let $0 < \kappa \le \dim X$ be a cardinal number.
Then $U$ has codimension $\kappa$ {{iff}} there exists a subspace $V$ of dimension $\kappa$ such that:
:$X = U \oplus V$
where $\oplus$ denote... | === Sufficient Condition ===
Suppose that $U$ has codimension $\kappa$.
Let $X/U$ be the quotient vector space of $X$ modulo $U$.
Since $U$ has codimension $\kappa$, we have $\map \dim {X/U} = \kappa$.
Let $q : X \to X/U$ be the quotient mapping.
Let $\BB_1 = \set {u_\alpha : \alpha \in I}$ be a basis for $U$, where $... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] such that $X \ne {\mathbf 0}_X$.
Let $U \subseteq X$ be a [[Definition:Vector Subspace|vector subspace]] of $X$.
Let $0 < \kappa \le \dim X$ be a [[Definition:Cardinal Number|cardinal number]].
Then $U... | === Sufficient Condition ===
Suppose that $U$ has [[Definition:Codimension of Vector Subspace|codimension]] $\kappa$.
Let $X/U$ be the [[Definition:Quotient Vector Space|quotient vector space of $X$ modulo $U$]].
Since $U$ has [[Definition:Codimension of Vector Subspace|codimension]] $\kappa$, we have $\map \dim {X/... | Characterization of Codimension in terms of Direct Sum | https://proofwiki.org/wiki/Characterization_of_Codimension_in_terms_of_Direct_Sum | https://proofwiki.org/wiki/Characterization_of_Codimension_in_terms_of_Direct_Sum | [
"Codimensions of Vector Subspaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Vector Subspace",
"Definition:Cardinal Number",
"Definition:Codimension of Vector Subspace",
"Definition:Vector Subspace",
"Definition:Dimension of Vector Space",
"Definition:Internal Direct Sum of Modules"
] | [
"Definition:Codimension of Vector Subspace",
"Definition:Quotient Vector Space",
"Definition:Codimension of Vector Subspace",
"Definition:Quotient Mapping",
"Definition:Basis of Vector Space",
"Definition:Indexing Set",
"Definition:Cardinality",
"Definition:Basis of Vector Space",
"Construction of B... |
proofwiki-22727 | Kernel of Non-Zero Linear Functional has Codimension 1 | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $f : X \to K$ be a non-zero linear functional
Let $\ker f$ be the kernel of $f$.
Then $\ker f$ has codimension $1$. | Let $X/\ker f$ be the quotient vector space of $X$ modulo $\ker f$.
Let $q : X \to X/\ker f$ be the quotient mapping.
From Condition for Mapping from Quotient Vector Space to be Well-Defined, there exists a linear functional $\widetilde f : X/\ker f \to K$ such that:
:$f = \widetilde f \circ q$
Note that if $\map {\wid... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $f : X \to K$ be a non-[[Definition:Zero Vector|zero]] [[Definition:Linear Functional|linear functional]]
Let $\ker f$ be the [[Definition:Kernel of Linear Transformation|kernel]] of $f$.
... | Let $X/\ker f$ be the [[Definition:Quotient Vector Space|quotient vector space of $X$ modulo $\ker f$]].
Let $q : X \to X/\ker f$ be the [[Definition:Quotient Mapping|quotient mapping]].
From [[Condition for Mapping from Quotient Vector Space to be Well-Defined]], there exists a [[Definition:Linear Functional|linear ... | Kernel of Non-Zero Linear Functional has Codimension 1 | https://proofwiki.org/wiki/Kernel_of_Non-Zero_Linear_Functional_has_Codimension_1 | https://proofwiki.org/wiki/Kernel_of_Non-Zero_Linear_Functional_has_Codimension_1 | [
"Codimensions of Vector Subspaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Zero Vector",
"Definition:Linear Functional",
"Definition:Kernel of Linear Transformation",
"Definition:Codimension of Vector Subspace"
] | [
"Definition:Quotient Vector Space",
"Definition:Quotient Mapping",
"Condition for Mapping from Quotient Vector Space to be Well-Defined",
"Definition:Linear Functional",
"Kernel of Quotient Mapping",
"Definition:Injection",
"Definition:Surjection",
"Linear Functional on Vector Space is Zero or Surject... |
proofwiki-22728 | Codimension of Intersection of Finitely Many Vector Subspaces is Bounded Above by Sum of Codimensions | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $U_1, U_2, \ldots, U_n$ be vector subspaces of $X$.
Let:
:$\ds U = \bigcap_{j \mathop = 1}^n U_j$
Then:
:$\ds \map {\operatorname {codim} } U \le \sum_{j \mathop = 1}^n \map {\operatorname {codim} } {U_j}$
where $\ds \sum_{j \mathop = 1}^n$ denotes the cardina... | Define the direct product:
:$\ds Y = \prod_{j \mathop = 1}^n \paren {X/U_j}$
For each $1 \le j \le n$, let $q_j : X \to X/U_j$ be the quotient mapping.
Define the linear transformation $T : X \to Y$ by:
:$T x = \tuple {\map {q_1} x, \ldots, \map {q_n} x}$ for each $x \in X$.
We have that:
:$\ds \bigcap_{j \mathop = 1}^... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $U_1, U_2, \ldots, U_n$ be [[Definition:Vector Subspace|vector subspaces]] of $X$.
Let:
:$\ds U = \bigcap_{j \mathop = 1}^n U_j$
Then:
:$\ds \map {\operatorname {codim} } U \le \sum_{j \m... | Define the [[Definition:Direct Product of Vector Spaces|direct product]]:
:$\ds Y = \prod_{j \mathop = 1}^n \paren {X/U_j}$
For each $1 \le j \le n$, let $q_j : X \to X/U_j$ be the [[Definition:Quotient Mapping|quotient mapping]].
Define the [[Definition:Linear Transformation|linear transformation]] $T : X \to Y$ by:... | Codimension of Intersection of Finitely Many Vector Subspaces is Bounded Above by Sum of Codimensions | https://proofwiki.org/wiki/Codimension_of_Intersection_of_Finitely_Many_Vector_Subspaces_is_Bounded_Above_by_Sum_of_Codimensions | https://proofwiki.org/wiki/Codimension_of_Intersection_of_Finitely_Many_Vector_Subspaces_is_Bounded_Above_by_Sum_of_Codimensions | [
"Codimensions of Vector Subspaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Vector Subspace",
"Definition:Sum of Cardinals"
] | [
"Definition:Direct Product of Vector Spaces",
"Definition:Quotient Mapping",
"Definition:Linear Transformation",
"Kernel of Quotient Mapping",
"Condition for Mapping from Quotient Vector Space to be Well-Defined",
"Definition:Linear Transformation",
"Definition:Injection",
"Kernel of Quotient Mapping"... |
proofwiki-22729 | Weakly Open Neighborhood of Zero Vector in Infinite-Dimensional Topological Vector Space contains Infinite-Dimensional Vector Subspace | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be an infinite-dimensional topological vector space over $\GF$.
Let $w$ be the weak topology of $\struct {X, \tau}$.
Let $U$ be an open neighborhood of ${\mathbf 0}_X$ in $\struct {X, w}$.
Then $U$ contains an infinite-dimensional vector subspace.
In particular, $U$... | Let $X^\ast$ be the topological dual space of $X$.
From Open Sets in Weak Topology of Topological Vector Space, there exists $f_1, \ldots, f_n \in X^\ast$ and $\epsilon > 0$ such that:
:$\set {x \in X : \cmod {\map {f_j} x} < \epsilon \text { for each } 1 \le j \le n} \subseteq U$
{{WLOG}} assume that $f_j \ne {\mathbf... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] of $\struct {X, \tau}$.
L... | Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $X$.
From [[Open Sets in Weak Topology of Topological Vector Space]], there exists $f_1, \ldots, f_n \in X^\ast$ and $\epsilon > 0$ such that:
:$\set {x \in X : \cmod {\map {f_j} x} < \epsilon \text { for each } 1 \le j \le n} \subsete... | Weakly Open Neighborhood of Zero Vector in Infinite-Dimensional Topological Vector Space contains Infinite-Dimensional Vector Subspace | https://proofwiki.org/wiki/Weakly_Open_Neighborhood_of_Zero_Vector_in_Infinite-Dimensional_Topological_Vector_Space_contains_Infinite-Dimensional_Vector_Subspace | https://proofwiki.org/wiki/Weakly_Open_Neighborhood_of_Zero_Vector_in_Infinite-Dimensional_Topological_Vector_Space_contains_Infinite-Dimensional_Vector_Subspace | [
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Infinite-Dimensional Vector Space",
"Definition:Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Open Neighborhood",
"Definition:Infinite-Dimensional Vector Space",
"Definition:Vector Subspace",
"Definition:Vector Subspace",
"Definition:Codimen... | [
"Definition:Topological Dual Space",
"Open Sets in Weak Topology of Topological Vector Space",
"Kernel of Linear Transformation is Linear Subspace",
"Definition:Vector Subspace",
"Set of Linear Subspaces is Closed under Intersection",
"Definition:Vector Subspace",
"Intersection of Finitely Many Kernels ... |
proofwiki-22730 | Infinite-Dimensional Banach Space contains Basic Sequence with Basis Constant Arbitrarily Close to 1 | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be an infinite-dimensional Banach space over $\GF$.
Then for each $\epsilon > 0$, $X$ contains a basic sequence with basis constant at most $1 + \epsilon$. | Let:
:$S_X = \set {x \in X : \norm x_X = 1}$
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\cl_w$ be the closure taken in $\struct {X, w}$.
From Weak Closure of Unit Sphere is Closed Unit Ball, we have ${\mathbf 0}_X \in \map {\cl_w} {S_X}$.
Further:
:$\ds \inf_{x \mathop \in S_X} \norm x ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Banach Space|Banach space]] over $\GF$.
Then for each $\epsilon > 0$, $X$ contains a [[Definition:Basic Sequence|basic sequence]] with [[Definition:Basis Cons... | Let:
:$S_X = \set {x \in X : \norm x_X = 1}$
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\cl_w$ be the [[Definition:Topological Closure|closure]] taken in $\struct {X, w}$.
From [[Weak Closure of Unit Sphere is Closed Unit Ball]]... | Infinite-Dimensional Banach Space contains Basic Sequence with Basis Constant Arbitrarily Close to 1 | https://proofwiki.org/wiki/Infinite-Dimensional_Banach_Space_contains_Basic_Sequence_with_Basis_Constant_Arbitrarily_Close_to_1 | https://proofwiki.org/wiki/Infinite-Dimensional_Banach_Space_contains_Basic_Sequence_with_Basis_Constant_Arbitrarily_Close_to_1 | [
"Basic Sequences"
] | [
"Definition:Infinite-Dimensional Vector Space",
"Definition:Banach Space",
"Definition:Basic Sequence",
"Definition:Basis Constant"
] | [
"Definition:Weak Topology on Topological Vector Space",
"Definition:Closure (Topology)",
"Weak Closure of Unit Sphere is Closed Unit Ball",
"Subset of Banach Space Bounded Away from Zero including Zero Vector in Weak Closure contains Basic Sequence with Basis Constant Arbitrarily Close to 1",
"Definition:Ba... |
proofwiki-22731 | Subset of Normed Vector Space is Bounded Away from Zero iff Closure does not contain Zero | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.
Let $S \subseteq X$.
Let $\cl$ be the closure in $\struct {X, \norm {\, \cdot \,} }$.
We have that:
:$\ds \inf_{x \mathop \in S} \norm x > 0$
{{iff}} ${\mathbf 0}_X \not \in \map \cl S$. | === Sufficient Condition ===
Suppose that ${\mathbf 0}_X \not \in \map \cl S$.
Then there exists an open neighborhood $U$ of ${\mathbf 0}_X$ such that $U \cap S = \O$.
From the definition of an open set in a normed vector space, there exists $M > 0$ such that:
:$\set {x \in X : \norm x_X < M} \subseteq U$
so that:
:$\s... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $S \subseteq X$.
Let $\cl$ be the [[Definition:Topological Closure|closure]] in $\struct {X, \norm {\, \cdot \,} }$.
We have that:
:$\ds \inf_{x \mathop \in S} \norm x > ... | === Sufficient Condition ===
Suppose that ${\mathbf 0}_X \not \in \map \cl S$.
Then there exists an [[Definition:Open Neighborhood|open neighborhood]] $U$ of ${\mathbf 0}_X$ such that $U \cap S = \O$.
From the definition of an [[Definition:Open Set/Normed Vector Space|open set in a normed vector space]], there exist... | Subset of Normed Vector Space is Bounded Away from Zero iff Closure does not contain Zero | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_is_Bounded_Away_from_Zero_iff_Closure_does_not_contain_Zero | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_is_Bounded_Away_from_Zero_iff_Closure_does_not_contain_Zero | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Closure (Topology)"
] | [
"Definition:Open Neighborhood",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Neighborhood"
] |
proofwiki-22732 | Outer Jordan Content of Degenerate Set | Let $M \subseteq \R^n$ be a bounded subset of Euclidean $n$-space.
Suppose that, for every $\tuple {x_1, \dots, x_n} \in M$, we have that:
:$x_k = \gamma$
for some fixed $k \in \set {1, \dots, n}$ and $\gamma \in \R$.
Then:
:$\map {m^*} M = 0$
where $m^*$ denotes the outer Jordan content. | For each $i \ne k$, let:
:$a_i$ be a lower bound of $\set {x_i : \tuple {x_1, \dots, x_n} \in M}$
:$b_i$ be an upper bound of $\set {x_i : \tuple {x_1, \dots, x_n} \in M}$
Consider the closed rectangle:
:$R = \closedint {a_1} {b_1} \times \dots \times \closedint {a_{k - 1}} {b_{k - 1}} \times \closedint \gamma \gamma \... | Let $M \subseteq \R^n$ be a [[Definition:Bounded Euclidean Space|bounded]] [[Definition:Subset|subset]] of [[Definition:Real Euclidean Space|Euclidean $n$-space]].
Suppose that, for every $\tuple {x_1, \dots, x_n} \in M$, we have that:
:$x_k = \gamma$
for some fixed $k \in \set {1, \dots, n}$ and $\gamma \in \R$.
Th... | For each $i \ne k$, let:
:$a_i$ be a [[Definition:Lower Bound of Subset of Real Numbers|lower bound]] of $\set {x_i : \tuple {x_1, \dots, x_n} \in M}$
:$b_i$ be an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] of $\set {x_i : \tuple {x_1, \dots, x_n} \in M}$
Consider the [[Definition:Closed Rectang... | Outer Jordan Content of Degenerate Set | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Degenerate_Set | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Degenerate_Set | [
"Outer Jordan Content"
] | [
"Definition:Bounded Metric Space/Euclidean",
"Definition:Subset",
"Definition:Euclidean Space/Real",
"Definition:Outer Jordan Content"
] | [
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Closed Rectangle",
"Definition:Cover of Set/Finite",
"Definition:Closed Rectangle",
"Category:Outer Jordan Content"
] |
proofwiki-22733 | Bounded Below Weakly Null Sequence in Infinite-Dimensional Banach Space contains Basic Subsequence with Basis Constant Arbitrarily Close to 1 | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be an infinite-dimensional Banach space over $\GF$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence converging to $x$ in $\struct {X, w}$ such that:
:$\ds \inf_{n \mathop \i... | Let $\cl_w$ be the closure taken in $\struct {X, w}$.
Let $S = \set {x_n : n \in \N}$.
From Point in Set Closure iff Limit of Net, we have ${\mathbf 0}_X \in \map {\cl_w} S$.
From Subset of Banach Space Bounded Away from Zero including Zero Vector in Weak Closure contains Basic Sequence with Basis Constant Arbitrarily ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Banach Space|Banach space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \... | Let $\cl_w$ be the [[Definition:Topological Closure|closure]] taken in $\struct {X, w}$.
Let $S = \set {x_n : n \in \N}$.
From [[Point in Set Closure iff Limit of Net]], we have ${\mathbf 0}_X \in \map {\cl_w} S$.
From [[Subset of Banach Space Bounded Away from Zero including Zero Vector in Weak Closure contains Bas... | Bounded Below Weakly Null Sequence in Infinite-Dimensional Banach Space contains Basic Subsequence with Basis Constant Arbitrarily Close to 1 | https://proofwiki.org/wiki/Bounded_Below_Weakly_Null_Sequence_in_Infinite-Dimensional_Banach_Space_contains_Basic_Subsequence_with_Basis_Constant_Arbitrarily_Close_to_1 | https://proofwiki.org/wiki/Bounded_Below_Weakly_Null_Sequence_in_Infinite-Dimensional_Banach_Space_contains_Basic_Subsequence_with_Basis_Constant_Arbitrarily_Close_to_1 | [
"Basic Sequences"
] | [
"Definition:Infinite-Dimensional Vector Space",
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Sequence",
"Definition:Convergent Sequence",
"Definition:Basic Sequence",
"Definition:Subsequence",
"Definition:Basis Constant"
] | [
"Definition:Closure (Topology)",
"Point in Set Closure iff Limit of Net",
"Subset of Banach Space Bounded Away from Zero including Zero Vector in Weak Closure contains Basic Sequence with Basis Constant Arbitrarily Close to 1",
"Definition:Basic Sequence",
"Definition:Basic Sequence",
"Definition:Basis Co... |
proofwiki-22734 | Outer Jordan Content of Subdivision | Let $\RR \subseteq \R^n$ be a closed $n$-rectangle.
Let $M \subseteq \RR$ be a subset.
Let $P$ be a finite subdivision of $\RR$.
Then:
:$\ds \map {m^*} M = \sum_{R \mathop \in P} \map {m^*} {M \cap R^\circ}$
where:
:$m^*$ denotes the outer Jordan content
:$R \in P$ denotes that $R$ is a subrectangle of $P$
:$R^\circ$ d... | Write $P = \tuple {P_1, P_2, \dots, P_n}$.
Further write $P_i = \set {x_{i,0}, \dots, x_{i, m_i}}$ for each $i$.
By applying Outer Jordan Content of Separated Union for each $\tuple {i, x_{i,j}}$, the result follows.
{{qed}}
Category:Outer Jordan Content
1vo5j7tteg6xz8dsp69u1hwo9qr232j | Let $\RR \subseteq \R^n$ be a [[Definition:Closed Rectangle|closed $n$-rectangle]].
Let $M \subseteq \RR$ be a [[Definition:Subset|subset]].
Let $P$ be a [[Definition:Finite Subdivision of Rectangle|finite subdivision]] of $\RR$.
Then:
:$\ds \map {m^*} M = \sum_{R \mathop \in P} \map {m^*} {M \cap R^\circ}$
where:
... | Write $P = \tuple {P_1, P_2, \dots, P_n}$.
Further write $P_i = \set {x_{i,0}, \dots, x_{i, m_i}}$ for each $i$.
By applying [[Outer Jordan Content of Separated Union]] for each $\tuple {i, x_{i,j}}$, the result follows.
{{qed}}
[[Category:Outer Jordan Content]]
1vo5j7tteg6xz8dsp69u1hwo9qr232j | Outer Jordan Content of Subdivision | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Subdivision | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Subdivision | [
"Outer Jordan Content"
] | [
"Definition:Closed Rectangle",
"Definition:Subset",
"Definition:Subdivision of Interval/Rectangle",
"Definition:Outer Jordan Content",
"Definition:Subdivision of Interval/Rectangle/Subrectangle",
"Definition:Interior (Topology)"
] | [
"Outer Jordan Content of Separated Union",
"Category:Outer Jordan Content"
] |
proofwiki-22735 | Sufficient Condition for Perturbation of Basic Sequence to be Basic Sequence | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a basic sequence such that there exists $f \in X^\ast$ with:
:... | Let $I_X : X \to X$ be the identity mapping.
From Existence of Distance Functional, there exists $\widetilde g \in X^\ast$ such that:
:$\norm {\widetilde g}_{X^\ast} = 1$
:$\ds \map {\widetilde g} u = \map {\operatorname {dist} } {u, \sqbrk {x_n}_{n \mathop \in \N} } := d$
:$\map {\widetilde g} x = 0$ for all $x \in \s... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {x_n}_{n \mathop \in \N}$ b... | Let $I_X : X \to X$ be the [[Definition:Identity Mapping|identity mapping]].
From [[Existence of Distance Functional]], there exists $\widetilde g \in X^\ast$ such that:
:$\norm {\widetilde g}_{X^\ast} = 1$
:$\ds \map {\widetilde g} u = \map {\operatorname {dist} } {u, \sqbrk {x_n}_{n \mathop \in \N} } := d$
:$\map {\... | Sufficient Condition for Perturbation of Basic Sequence to be Basic Sequence | https://proofwiki.org/wiki/Sufficient_Condition_for_Perturbation_of_Basic_Sequence_to_be_Basic_Sequence | https://proofwiki.org/wiki/Sufficient_Condition_for_Perturbation_of_Basic_Sequence_to_be_Basic_Sequence | [
"Basic Sequences"
] | [
"Definition:Banach Space",
"Definition:Normed Dual Space",
"Definition:Basic Sequence",
"Definition:Closed Linear Span",
"Definition:Basic Sequence"
] | [
"Definition:Identity Mapping",
"Existence of Distance Functional",
"Normed Dual Space is Banach Space",
"Fundamental Property of Norm on Bounded Linear Functional",
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation",
"Space of Bounded Linear Transformations forms Vect... |
proofwiki-22736 | Outer Jordan Content of Closed Rectangle | Let $\RR \subseteq \R^n$ be a closed $n$-rectangle.
Then:
:$\map {m^*} \RR = \map V \RR$
where:
:$m^*$ denotes the outer Jordan content
:$V$ denotes the content of a rectangle | Clearly, $\set \RR$ is a finite covering of $\RR$ by closed $n$-rectangles.
Therefore:
:$\map {m^*} \RR \le \map V \RR$
{{qed|lemma}}
Now, let $\varepsilon > 0$ be arbitrary.
By Characterizing Property of Infimum of Subset of Real Numbers, there is a finite covering $\CC$ of $\RR$ by closed $n$-rectangles such that:
:$... | Let $\RR \subseteq \R^n$ be a [[Definition:Closed Rectangle|closed $n$-rectangle]].
Then:
:$\map {m^*} \RR = \map V \RR$
where:
:$m^*$ denotes the [[Definition:Outer Jordan Content|outer Jordan content]]
:$V$ denotes the [[Definition:Content of Rectangle|content of a rectangle]] | Clearly, $\set \RR$ is a [[Definition:Finite Cover|finite covering]] of $\RR$ by [[Definition:Closed Rectangle|closed $n$-rectangles]].
Therefore:
:$\map {m^*} \RR \le \map V \RR$
{{qed|lemma}}
Now, let $\varepsilon > 0$ be arbitrary.
By [[Characterizing Property of Infimum of Subset of Real Numbers]], there is a [... | Outer Jordan Content of Closed Rectangle | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Closed_Rectangle | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Closed_Rectangle | [
"Outer Jordan Content"
] | [
"Definition:Closed Rectangle",
"Definition:Outer Jordan Content",
"Definition:Content of Rectangle"
] | [
"Definition:Cover of Set/Finite",
"Definition:Closed Rectangle",
"Characterizing Property of Infimum of Subset of Real Numbers",
"Definition:Cover of Set/Finite",
"Definition:Closed Rectangle",
"Definition:Subdivision of Interval/Finite",
"Definition:Interval/Ordered Set/Endpoint",
"Definition:Subdivi... |
proofwiki-22737 | Terms in Convergent Series Converge to Zero/Normed Vector Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $\sequence {x_j}_{j \mathop \in \N}$ be a sequence such that:
:$\ds \sum_{j \mathop = 1}^\infty x_j$ converges.
Then:
:$x_j \to {\mathbf 0}_X$ in $\struct {X, \norm {\, \cdot \,} }$ as $j \to \infty$. | Let:
:$\ds s_n = \sum_{j \mathop = 1}^n x_j$
Let $\epsilon > 0$.
Since $\sequence {s_n}_{n \mathop \in \N}$ converges, it is Cauchy by Convergent Sequence is Cauchy Sequence.
Hence there exists $N \in \N$ such that for $m, n \ge N$ we have:
:$\norm {s_m - s_n}_X < \epsilon$
Then in particular:
:$\norm {s_n - s_{n - 1}... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\sequence {x_j}_{j \mathop \in \N}$ be a [[Definition:Sequence|sequence]] such that:
:$\ds \sum_{j \mathop = 1}^\infty x_j$ [[Definition:Convergent Series|converges]].
T... | Let:
:$\ds s_n = \sum_{j \mathop = 1}^n x_j$
Let $\epsilon > 0$.
Since $\sequence {s_n}_{n \mathop \in \N}$ [[Definition:Convergent Sequence|converges]], it is [[Definition:Cauchy Sequence|Cauchy]] by [[Convergent Sequence is Cauchy Sequence]].
Hence there exists $N \in \N$ such that for $m, n \ge N$ we have:
:$\no... | Terms in Convergent Series Converge to Zero/Normed Vector Space | https://proofwiki.org/wiki/Terms_in_Convergent_Series_Converge_to_Zero/Normed_Vector_Space | https://proofwiki.org/wiki/Terms_in_Convergent_Series_Converge_to_Zero/Normed_Vector_Space | [
"Terms in Convergent Series Converge to Zero"
] | [
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Convergent Series"
] | [
"Definition:Convergent Sequence",
"Definition:Cauchy Sequence",
"Convergent Sequence is Cauchy Sequence",
"Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence",
"Category:Terms in Convergent Series Converge to Zero"
] |
proofwiki-22738 | Point in Set Closure iff Limit of Filter | Let $\struct {X, \tau}$ be a topological space.
Let $\cl$ denote closure in $\struct {X, \tau}$.
Let $\FF$ be a filter on $X$.
Let $E \subseteq X$.
Let $x \in X$.
We have that $x \in \map \cl E$ {{iff}} there exists a filter on $X$ such that:
:$\FF$ converges to $x$
and:
:$E \in \FF$ | === Necessary Condition ===
Let $\UU_x$ be the neighborhood filter at $x$.
Suppose that $x \in \map \cl E$.
Then $U \cap E \ne \O$ for all $U \in \UU_x$.
Let $\BB = \set {U \cap E : U \in \UU_x}$.
Since $X \in \UU_x$, we have $E \in \BB$ and hence $\BB \ne \O$.
Since $x \in \map \cl E$, we have that:
:$U \cap E \ne \O... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\cl$ denote [[Definition:Topological Closure|closure]] in $\struct {X, \tau}$.
Let $\FF$ be a [[Definition:Filter on Set|filter]] on $X$.
Let $E \subseteq X$.
Let $x \in X$.
We have that $x \in \map \cl E$ {{iff}} there exists ... | === Necessary Condition ===
Let $\UU_x$ be the [[Definition:Neighborhood Filter|neighborhood filter]] at $x$.
Suppose that $x \in \map \cl E$.
Then $U \cap E \ne \O$ for all $U \in \UU_x$.
Let $\BB = \set {U \cap E : U \in \UU_x}$.
Since $X \in \UU_x$, we have $E \in \BB$ and hence $\BB \ne \O$.
Since $x \in \ma... | Point in Set Closure iff Limit of Filter | https://proofwiki.org/wiki/Point_in_Set_Closure_iff_Limit_of_Filter | https://proofwiki.org/wiki/Point_in_Set_Closure_iff_Limit_of_Filter | [
"Limits of Filters",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Closure (Topology)",
"Definition:Filter on Set",
"Definition:Filter on Set",
"Definition:Convergent Filter"
] | [
"Definition:Neighborhood Filter",
"Definition:Closure (Topology)",
"Definition:Filter Basis/Generated Filter",
"Definition:Filter on Set",
"Definition:Convergent Filter",
"Definition:Filter on Set",
"Definition:Convergent Filter",
"Definition:Convergent Filter"
] |
proofwiki-22739 | Filter Generated by Net is Well-Defined | Let $X$ be a set.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a net.
Then the filter generated by $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ is well-defined. | For each $\lambda_0 \in \Lambda$, let $B_{\lambda_0} = \set {x_\lambda : \lambda \succeq \lambda_0}$.
Let $\BB = \set {B_\lambda : \lambda \in \Lambda}$.
Clearly $\BB \ne \O$ and $\O \not \in \BB$.
We need to show that for each $\lambda_1, \lambda_2 \in \Lambda$, there exists $\lambda_0 \in \Lambda$ such that $B_{\lam... | Let $X$ be a [[Definition:Set|set]].
Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a [[Definition:Net (Set Theory)|net]].
Then the [[Definition:Filter Generated by Net|filter generated by $\family {x_\lambda}_{\lambda \math... | For each $\lambda_0 \in \Lambda$, let $B_{\lambda_0} = \set {x_\lambda : \lambda \succeq \lambda_0}$.
Let $\BB = \set {B_\lambda : \lambda \in \Lambda}$.
Clearly $\BB \ne \O$ and $\O \not \in \BB$.
We need to show that for each $\lambda_1, \lambda_2 \in \Lambda$, there exists $\lambda_0 \in \Lambda$ such that $B_{\... | Filter Generated by Net is Well-Defined | https://proofwiki.org/wiki/Filter_Generated_by_Net_is_Well-Defined | https://proofwiki.org/wiki/Filter_Generated_by_Net_is_Well-Defined | [
"Filters on Sets"
] | [
"Definition:Set",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Filter Generated by Net"
] | [
"Definition:Directed Preordering",
"Definition:Transitive Relation",
"Category:Filters on Sets"
] |
proofwiki-22740 | Filter Generated by Net Converges iff Net Converges | Let $\struct {X, \tau}$ be a topological space.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a net.
Let $\FF$ be the filter generated by $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$.
Let $x \in X$.
Then $\FF$ converges to $x$ {{iff}} $\family {x_\... | For each $\lambda_0 \in \Lambda$, set:
:$B_{\lambda_0} = \set {x_\lambda : \lambda_0 \succeq \lambda}$
Let $\UU_x$ be the neighborhood filter at $x$.
We have that $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ converges to $x$ {{iff}} for each $U \in \UU_x$, there exists $\lambda_0 \in \Lambda$ such that $B_{\lam... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a [[Definition:Net (Set Theory)|net]].
Let $\FF$ be the [[Definition:Filter Generated by Net|filt... | For each $\lambda_0 \in \Lambda$, set:
:$B_{\lambda_0} = \set {x_\lambda : \lambda_0 \succeq \lambda}$
Let $\UU_x$ be the [[Definition:Neighborhood Filter|neighborhood filter]] at $x$.
We have that $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ [[Definition:Convergent Net|converges]] to $x$ {{iff}} for each $U ... | Filter Generated by Net Converges iff Net Converges | https://proofwiki.org/wiki/Filter_Generated_by_Net_Converges_iff_Net_Converges | https://proofwiki.org/wiki/Filter_Generated_by_Net_Converges_iff_Net_Converges | [
"Filters Generated by Nets"
] | [
"Definition:Topological Space",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Filter Generated by Net",
"Definition:Convergent Filter",
"Definition:Convergent Net"
] | [
"Definition:Neighborhood Filter",
"Definition:Convergent Net",
"Definition:Convergent Net"
] |
proofwiki-22741 | Outer Jordan Content of Open Rectangle | Let:
:$\ds R = \prod_{i \mathop = 1}^n \openint {a_i} {b_i}$
be an open $n$-rectangle, where:
:$a_i \le b_i$
for each $i \in \set {1, \dots, n}$.
Then:
:$\ds \map {m^*} R = \prod_{i \mathop = 1}^n \paren {b_i - a_i}$
where $m^*$ denotes the outer Jordan content. | By Outer Jordan Content of Closed Rectangle, we know that, for:
:$\ds \overline R = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$
the outer Jordan content is:
:$\ds \map {m^*} {\overline R} = \map V {\overline R} = \prod_{i \mathop = 1}^n \paren {b_i} {a_i}$
Let us form the subdivision:
:$P = \tuple {P_1, \dots, P_n}... | Let:
:$\ds R = \prod_{i \mathop = 1}^n \openint {a_i} {b_i}$
be an [[Definition:Open Rectangle|open $n$-rectangle]], where:
:$a_i \le b_i$
for each $i \in \set {1, \dots, n}$.
Then:
:$\ds \map {m^*} R = \prod_{i \mathop = 1}^n \paren {b_i - a_i}$
where $m^*$ denotes the [[Definition:Outer Jordan Content|outer Jordan ... | By [[Outer Jordan Content of Closed Rectangle]], we know that, for:
:$\ds \overline R = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$
the [[Definition:Outer Jordan Content|outer Jordan content]] is:
:$\ds \map {m^*} {\overline R} = \map V {\overline R} = \prod_{i \mathop = 1}^n \paren {b_i} {a_i}$
Let us form the [... | Outer Jordan Content of Open Rectangle | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Open_Rectangle | https://proofwiki.org/wiki/Outer_Jordan_Content_of_Open_Rectangle | [
"Outer Jordan Content"
] | [
"Definition:Open Rectangle",
"Definition:Outer Jordan Content"
] | [
"Outer Jordan Content of Closed Rectangle",
"Definition:Outer Jordan Content",
"Definition:Subdivision of Interval/Rectangle",
"Outer Jordan Content of Subdivision",
"Category:Outer Jordan Content"
] |
proofwiki-22742 | Point is Cluster Point of Net iff Limit of Subnet | Let $\struct {X, \tau}$ be a topological space.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a net.
Let $x \in X$.
Then $x$ is a cluster point of $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ {{iff}}:
:there exists a directed set $\struct {A, \sqsu... | For each $x \in X$, let $\UU_x$ be the neighborhood filter at $x$. | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a [[Definition:Net|net]].
Let $x \in X$.
Then $x$ is a [[Definition:Convergent Net/Cluster Poin... | For each $x \in X$, let $\UU_x$ be the [[Definition:Neighborhood Filter|neighborhood filter]] at $x$. | Point is Cluster Point of Net iff Limit of Subnet | https://proofwiki.org/wiki/Point_is_Cluster_Point_of_Net_iff_Limit_of_Subnet | https://proofwiki.org/wiki/Point_is_Cluster_Point_of_Net_iff_Limit_of_Subnet | [
"Nets (Set Theory)"
] | [
"Definition:Topological Space",
"Definition:Directed Preordering",
"Definition:Net",
"Definition:Convergent Net/Cluster Point",
"Definition:Directed Preordering",
"Definition:Increasing/Mapping",
"Definition:Cofinal Mapping",
"Definition:Subnet",
"Definition:Convergent Net"
] | [
"Definition:Neighborhood Filter"
] |
proofwiki-22743 | Product of Preordered Sets is Preordered | Let $\struct {P_1, \preceq_1}$ and $\struct {P_2, \preceq_2}$ be preordered sets.
Let $P = P_1 \times P_2$.
Define the relation $\preceq$ on $P$ by:
:$\tuple {p_1, p_2} \preceq \tuple {q_1, q_2}$
for each $\tuple {p_1, p_2}, \tuple {q_1, q_2} \in P$.
{{Explain|What type of preordering is $\preceq$? Create and link to ... | === $\preceq$ is reflexive ===
Let $\tuple {p_1, p_2} \in P$.
Since $\preceq_1$ and $\preceq_2$ are preorderings, we have $p_1 \preceq_1 p_1$ and $p_2 \preceq p_2$.
Hence $\tuple {p_1, p_2} \preceq \tuple {p_1, p_2}$.
Hence $\preceq$ is reflexive.
{{qed|lemma}} | Let $\struct {P_1, \preceq_1}$ and $\struct {P_2, \preceq_2}$ be [[Definition:Preordered Set|preordered sets]].
Let $P = P_1 \times P_2$.
Define the [[Definition:Relation|relation]] $\preceq$ on $P$ by:
:$\tuple {p_1, p_2} \preceq \tuple {q_1, q_2}$
for each $\tuple {p_1, p_2}, \tuple {q_1, q_2} \in P$.
{{Explain|Wh... | === $\preceq$ is [[Definition:Reflexive Relation|reflexive]] ===
Let $\tuple {p_1, p_2} \in P$.
Since $\preceq_1$ and $\preceq_2$ are [[Definition:Preordering|preorderings]], we have $p_1 \preceq_1 p_1$ and $p_2 \preceq p_2$.
Hence $\tuple {p_1, p_2} \preceq \tuple {p_1, p_2}$.
Hence $\preceq$ is [[Definition:Refle... | Product of Preordered Sets is Preordered | https://proofwiki.org/wiki/Product_of_Preordered_Sets_is_Preordered | https://proofwiki.org/wiki/Product_of_Preordered_Sets_is_Preordered | [
"Preorderings"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Relation",
"Definition:Order Product",
"Definition:Preordering"
] | [
"Definition:Reflexive Relation",
"Definition:Preordering",
"Definition:Reflexive Relation"
] |
proofwiki-22744 | Characterization of Compactness in terms of Nets | Let $\struct {X, \tau}$ be a topological space.
Then $X$ is compact {{iff}} every net in $X$ has a convergent subnet. | For each $x \in X$, let $\UU_x$ be the neighborhood filter at $x$. | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Then $X$ is [[Definition:Compact Topological Space|compact]] {{iff}} every [[Definition:Net (Set Theory)|net]] in $X$ has a [[Definition:Convergent Net|convergent]] [[Definition:Subnet|subnet]]. | For each $x \in X$, let $\UU_x$ be the [[Definition:Neighborhood Filter|neighborhood filter]] at $x$. | Characterization of Compactness in terms of Nets | https://proofwiki.org/wiki/Characterization_of_Compactness_in_terms_of_Nets | https://proofwiki.org/wiki/Characterization_of_Compactness_in_terms_of_Nets | [
"Characterization of Compactness in terms of Nets",
"Compact Topological Spaces",
"Nets (Set Theory)"
] | [
"Definition:Topological Space",
"Definition:Compact Topological Space",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Subnet"
] | [
"Definition:Neighborhood Filter"
] |
proofwiki-22745 | Cluster Point of Convergent Net in Hausdorff Space is Limit | Let $\struct {X, \tau}$ be a Hausdorff space.
Let $x \in X$.
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a convergent net with limit $x$.
Let $y$ be a cluster point of $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$.
Then $x = y$. | {{AimForCont}} that $x \ne y$.
Since $\struct {X, \tau}$ is Hausdorff space, there exists an open neighborhood $U$ of $x$ and an open neighborhood $v$ of $y$ such that:
:$U \cap V = \O$
Since $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ converges to $x$, there exists $\lambda_0 \in \Lambda$ such that:
:$x_\lam... | Let $\struct {X, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $x \in X$.
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a [[Definition:Convergent Net|convergent net]] with [[Definition:Limit of Net|limit]] $x$.
Let $y$ be a [[Definition:Convergent Net/Cluster Point|cluster point]] of $... | {{AimForCont}} that $x \ne y$.
Since $\struct {X, \tau}$ is [[Definition:Hausdorff Space|Hausdorff space]], there exists an [[Definition:Open Neighborhood|open neighborhood]] $U$ of $x$ and an [[Definition:Open Neighborhood|open neighborhood]] $v$ of $y$ such that:
:$U \cap V = \O$
Since $\family {x_\lambda}_{\lamb... | Cluster Point of Convergent Net in Hausdorff Space is Limit | https://proofwiki.org/wiki/Cluster_Point_of_Convergent_Net_in_Hausdorff_Space_is_Limit | https://proofwiki.org/wiki/Cluster_Point_of_Convergent_Net_in_Hausdorff_Space_is_Limit | [
"Convergent Nets",
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Definition:Convergent Net/Cluster Point"
] | [
"Definition:T2 Space",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Convergent Net",
"Definition:Convergent Net/Cluster Point",
"Category:Convergent Nets",
"Category:Hausdorff Spaces"
] |
proofwiki-22746 | Weak Cluster Point of Basic Sequence in Banach Space is Zero Vector | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a basic sequence.
Let $x$ be a cluster point of $\sequence {x_n}_{n \mathop \in \N}$ in $\struct {X, w}$.
... | Let $\sqbrk {x_n}_{n \mathop \in \N}$ be the closed linear span of $\sequence {x_n}_{n \mathop \in \N}$.
Let $\cl_w$ be the closure taken in $\struct {X, w}$.
Since $x$ is a cluster point of $\sequence {x_n}_{n \mathop \in \N}$ in $\struct {X, w}$:
:there exists a subnet $\family {x_{n_\lambda} }_{\lambda \mathop \in ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Bas... | Let $\sqbrk {x_n}_{n \mathop \in \N}$ be the [[Definition:Closed Linear Span|closed linear span]] of $\sequence {x_n}_{n \mathop \in \N}$.
Let $\cl_w$ be the [[Definition:Topological Closure|closure]] taken in $\struct {X, w}$.
Since $x$ is a [[Definition:Convergent Net/Cluster Point|cluster point]] of $\sequence {x... | Weak Cluster Point of Basic Sequence in Banach Space is Zero Vector | https://proofwiki.org/wiki/Weak_Cluster_Point_of_Basic_Sequence_in_Banach_Space_is_Zero_Vector | https://proofwiki.org/wiki/Weak_Cluster_Point_of_Basic_Sequence_in_Banach_Space_is_Zero_Vector | [
"Basic Sequences"
] | [
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Basic Sequence",
"Definition:Convergent Net/Cluster Point"
] | [
"Definition:Closed Linear Span",
"Definition:Closure (Topology)",
"Definition:Convergent Net/Cluster Point",
"Definition:Subnet",
"Definition:Convergent Net",
"Point is Cluster Point of Net iff Limit of Subnet",
"Point in Set Closure iff Limit of Net",
"Linear Subspace is Convex Set",
"Definition:Co... |
proofwiki-22747 | Characterization of Compactness in terms of Nets/Corollary | :$X$ is compact {{iff}} every net in $X$ has a cluster point. | From Characterization of Compactness in terms of Nets, we have that:
:$X$ is compact {{iff}} every net in $X$ has a convergent subnet.
From Point is Cluster Point of Net iff Limit of Subnet:
:a point $x$ is a cluster point of a net {{iff}} it is the limit of a convergent subnet.
Hence:
:$X$ compact {{iff}} every net in... | :$X$ is [[Definition:Compact Topological Space|compact]] {{iff}} every [[Definition:Net (Set Theory)|net]] in $X$ has a [[Definition:Convergent Net/Cluster Point|cluster point]]. | From [[Characterization of Compactness in terms of Nets]], we have that:
:$X$ is [[Definition:Compact Topological Space|compact]] {{iff}} every [[Definition:Net (Set Theory)|net]] in $X$ has a [[Definition:Convergent Net|convergent]] [[Definition:Subnet|subnet]].
From [[Point is Cluster Point of Net iff Limit of Subne... | Characterization of Compactness in terms of Nets/Corollary | https://proofwiki.org/wiki/Characterization_of_Compactness_in_terms_of_Nets/Corollary | https://proofwiki.org/wiki/Characterization_of_Compactness_in_terms_of_Nets/Corollary | [
"Characterization of Compactness in terms of Nets"
] | [
"Definition:Compact Topological Space",
"Definition:Net (Set Theory)",
"Definition:Convergent Net/Cluster Point"
] | [
"Characterization of Compactness in terms of Nets",
"Definition:Compact Topological Space",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Subnet",
"Point is Cluster Point of Net iff Limit of Subnet",
"Definition:Cluster Point",
"Definition:Net (Set Theory)",
"Definition:Lim... |
proofwiki-22748 | Kadets-Pełczyński Criterion on the Existence of a Basic Sequence | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\cl_w$ be the closure in $\struct {X, w}$.
Let $S \subseteq X$ be such that:
:$\ds 0 < \inf_{x \mathop \in S} \norm x_X \le \sup_{x \mathop \in S... | === $(2)$ implies $(1)$ ===
{{AimForCont}} that $S$ contains a basic sequence $\sequence {x_n}_{n \mathop \in \N}$.
Since $\map {\cl_w} S$ is compact in $w$, $\sequence {x_n}_{n \mathop \in \N}$ has a cluster point $x \in \map {\cl_w} S$ in $\struct {X, w}$ by {{Corollary|Characterization of Compactness in terms of Net... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\cl_w$ be the [[Definition:Topological Closure|closure]] ... | === $(2)$ implies $(1)$ ===
{{AimForCont}} that $S$ contains a [[Definition:Basic Sequence|basic sequence]] $\sequence {x_n}_{n \mathop \in \N}$.
Since $\map {\cl_w} S$ is [[Definition:Compact Topological Space|compact]] in $w$, $\sequence {x_n}_{n \mathop \in \N}$ has a [[Definition:Convergent Net|cluster point]] $x... | Kadets-Pełczyński Criterion on the Existence of a Basic Sequence | https://proofwiki.org/wiki/Kadets-Pełczyński_Criterion_on_the_Existence_of_a_Basic_Sequence | https://proofwiki.org/wiki/Kadets-Pełczyński_Criterion_on_the_Existence_of_a_Basic_Sequence | [
"Basic Sequences"
] | [
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Closure (Topology)",
"Definition:Basic Sequence",
"Definition:Compact Topological Space"
] | [
"Definition:Basic Sequence",
"Definition:Compact Topological Space",
"Definition:Convergent Net",
"Weak Cluster Point of Basic Sequence in Banach Space is Zero Vector",
"Definition:Basic Sequence",
"Definition:Basic Sequence",
"Definition:Basic Sequence",
"Definition:Compact Topological Space",
"Def... |
proofwiki-22749 | Subsequence of Basic Sequence in Banach Space is Basic Sequence | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a basic sequence in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {x_{n_k} }_{k \mathop \in \N}$ be a subsequence of $\sequence {x_n}_{n \mathop \in \N}$.
Then $\sequenc... | By Grunblum's Criterion, there exists $K > 0$ such that for any sequence $\sequence {\alpha_j}_{j \mathop \in \N}$ in $\GF$ such that:
:$\ds \norm {\sum_{j \mathop = 1}^m \alpha_j x_j}_X \le K \norm {\sum_{j \mathop = 1}^n \alpha_j x_j}_X$ for $m \le n$.
In particular, we have:
:$\ds \norm {\sum_{j \mathop = 1}^{n_k} \... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Basic Sequence|basic sequence]] in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {x_{n_k} }_{k \mathop \in \N}$ be a [[Defi... | By [[Grunblum's Criterion]], there exists $K > 0$ such that for any [[Definition:Sequence|sequence]] $\sequence {\alpha_j}_{j \mathop \in \N}$ in $\GF$ such that:
:$\ds \norm {\sum_{j \mathop = 1}^m \alpha_j x_j}_X \le K \norm {\sum_{j \mathop = 1}^n \alpha_j x_j}_X$ for $m \le n$.
In particular, we have:
:$\ds \norm ... | Subsequence of Basic Sequence in Banach Space is Basic Sequence/Proof 2 | https://proofwiki.org/wiki/Subsequence_of_Basic_Sequence_in_Banach_Space_is_Basic_Sequence | https://proofwiki.org/wiki/Subsequence_of_Basic_Sequence_in_Banach_Space_is_Basic_Sequence/Proof_2 | [
"Basic Sequences",
"Subsequence of Basic Sequence in Banach Space is Basic Sequence"
] | [
"Definition:Banach Space",
"Definition:Basic Sequence",
"Definition:Subsequence",
"Definition:Basic Sequence"
] | [
"Grunblum's Criterion",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Basic Sequence",
"Grunblum's Criterion"
] |
proofwiki-22750 | Unique Accumulation Point of Sequence in Weakly Countably Compact Subset of Banach Space is Limit of Sequence | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $A \subseteq X$ be countably compact in $\struct {X, w}$.
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $A$ such that:
... | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Suppose that $\sequence {x_n}_{n \mathop \in \N}$ does not converge to $x$.
By the definition of weak convergence, there exists $f \in X^\ast$ such that:
:$\map f {x_n} \not \to \map f x$
Hence there ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $A \subseteq X$ be [[Definition:Countably Compact Space|co... | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Suppose that $\sequence {x_n}_{n \mathop \in \N}$ does not [[Definition:Convergent Sequence|converge]] to $x$.
By the definition of [[Definition:Weak Convergence|wea... | Unique Accumulation Point of Sequence in Weakly Countably Compact Subset of Banach Space is Limit of Sequence | https://proofwiki.org/wiki/Unique_Accumulation_Point_of_Sequence_in_Weakly_Countably_Compact_Subset_of_Banach_Space_is_Limit_of_Sequence | https://proofwiki.org/wiki/Unique_Accumulation_Point_of_Sequence_in_Weakly_Countably_Compact_Subset_of_Banach_Space_is_Limit_of_Sequence | [
"Countably Compact Spaces",
"Weak Topologies on Topological Vector Spaces",
"Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Countably Compact Space",
"Definition:Sequence",
"Definition:Accumulation Point/Sequence",
"Definition:Weak Convergence"
] | [
"Definition:Normed Dual Space",
"Definition:Convergent Sequence",
"Definition:Weak Convergence",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius",
"Definition:Accumulation Point/Sequence",
"Definition:Countably Compact Space",
"Definition:Accumulation Point/Sequen... |
proofwiki-22751 | Eberlein-Šmulian Theorem | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $A \subseteq X$.
{{TFAE}}
:$(1): \quad$ $\struct {A, w}$ is compact
:$(2): \quad$ $\struct {A, w}$ is sequentially compact in itself
:$(3): \quad$... | Let $\cl_w$ be the closure taken in $\struct {X, w}$.
Let $\cl$ be the closure taken in $\struct {X, \norm {\, \cdot \,}_X}$. | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $A \subseteq X$.
{{TFAE}}
:$(1): \quad$ $\struct {A, w}$... | Let $\cl_w$ be the [[Definition:Topological Closure|closure]] taken in $\struct {X, w}$.
Let $\cl$ be the [[Definition:Topological Closure|closure]] taken in $\struct {X, \norm {\, \cdot \,}_X}$. | Eberlein-Šmulian Theorem | https://proofwiki.org/wiki/Eberlein-Šmulian_Theorem | https://proofwiki.org/wiki/Eberlein-Šmulian_Theorem | [
"Eberlein-Šmulian Theorem",
"Weak Topologies on Topological Vector Spaces",
"Banach Spaces",
"Compact Topological Spaces",
"Sequentially Compact Spaces",
"Countably Compact Spaces"
] | [
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Compact Topological Space",
"Definition:Sequentially Compact Space/In Itself",
"Definition:Countably Compact Space"
] | [
"Definition:Closure (Topology)",
"Definition:Closure (Topology)",
"Definition:Closure (Topology)"
] |
proofwiki-22752 | Countably Compact Metric Space is Bounded | Let $\struct {X, d}$ be a countably compact metric space.
Then $\struct {X, d}$ is bounded. | Let $x_0 \in X$.
Let $\map {B_n} {x_0}$ be the open ball of radius $n$, centered at $x_0$.
Clearly:
:$\ds \bigcup_{n \mathop = 1}^\infty \map {B_n} {x_0} \subseteq X$
For each $x \in X$, we have:
:$x \in \map {B_r} {x_0}$, where $r = \floor {\map d {x, x_0} } + 1$.
Hence, we have:
:$\ds X = \bigcup_{n \mathop = 1}^\in... | Let $\struct {X, d}$ be a [[Definition:Countably Compact Space|countably compact]] [[Definition:Metric Space|metric space]].
Then $\struct {X, d}$ is [[Definition:Bounded Metric Space|bounded]]. | Let $x_0 \in X$.
Let $\map {B_n} {x_0}$ be the [[Definition:Open Ball|open ball]] of [[Definition:Radius of Open Ball|radius]] $n$, [[Definition:Center of Open Ball|centered]] at $x_0$.
Clearly:
:$\ds \bigcup_{n \mathop = 1}^\infty \map {B_n} {x_0} \subseteq X$
For each $x \in X$, we have:
:$x \in \map {B_r} {x_0}$... | Countably Compact Metric Space is Bounded | https://proofwiki.org/wiki/Countably_Compact_Metric_Space_is_Bounded | https://proofwiki.org/wiki/Countably_Compact_Metric_Space_is_Bounded | [
"Countably Compact Spaces",
"Metric Spaces"
] | [
"Definition:Countably Compact Space",
"Definition:Metric Space",
"Definition:Bounded Metric Space"
] | [
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Definition:Countably Compact Space",
"Definition:Bounded Metric Space",
"Category:Countably Compact Spaces",
"Category:Metric Spaces"
] |
proofwiki-22753 | Continuous Image of Countably Compact Space is Countably Compact | Let $\struct {X, \tau_X}$, $\struct {Y, \tau_X}$ be topological spaces.
Let $C \subseteq X$ be countably compact in $\struct {X, \tau_X}$.
Let $f : X \to Y$ be continuous.
Then $f \sqbrk C$ is countably compact. | Let $\UU = \set {U_n : n \in \N}$ be a countable open cover of $f \sqbrk C$.
Since $f$ is continuous, we have $f^{-1} \sqbrk {U_n} \in \tau_X$ for each $n \in \N$.
{{begin-eqn}}
{{eqn | l = \bigcup_{n \mathop = 1}^\infty f^{-1} \sqbrk {U_n}
| r = f^{-1} \sqbrk {\bigcup_{n \mathop = 1}^\infty U_n}
| c = Preimage of... | Let $\struct {X, \tau_X}$, $\struct {Y, \tau_X}$ be [[Definition:Topological Space|topological spaces]].
Let $C \subseteq X$ be [[Definition:Countably Compact Space|countably compact]] in $\struct {X, \tau_X}$.
Let $f : X \to Y$ be [[Definition:Continuous Mapping|continuous]].
Then $f \sqbrk C$ is [[Definition:Coun... | Let $\UU = \set {U_n : n \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Open Cover|open cover]] of $f \sqbrk C$.
Since $f$ is [[Definition:Continuous Mapping|continuous]], we have $f^{-1} \sqbrk {U_n} \in \tau_X$ for each $n \in \N$.
{{begin-eqn}}
{{eqn | l = \bigcup_{n \mathop = 1}^\infty f^{-1} ... | Continuous Image of Countably Compact Space is Countably Compact | https://proofwiki.org/wiki/Continuous_Image_of_Countably_Compact_Space_is_Countably_Compact | https://proofwiki.org/wiki/Continuous_Image_of_Countably_Compact_Space_is_Countably_Compact | [
"Countably Compact Spaces"
] | [
"Definition:Topological Space",
"Definition:Countably Compact Space",
"Definition:Continuous Mapping",
"Definition:Countably Compact Space"
] | [
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Continuous Mapping",
"Preimage of Union under Mapping/General Result",
"Image of Preimage under Mapping",
"Definition:Open Cover",
"Definition:Countably Compact Space",
"Definition:Subcover/Finite",
"Definition:Countably Compact Space"... |
proofwiki-22754 | Subset of Normed Vector Space providing Bounded Point Evaluations is Norm Bounded | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $S \subseteq X$.
Suppose that:
:$\ds \sup_{x \mathop \in S} \cmod {\map f x} < \infty$
for each... | Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\iota : X \to X^{\ast \ast}$ be the evaluation linear transformation.
We then have:
:$\ds \sup_{x \mathop \in S} \cmod {\map {\iota x} f} < \infty$ for each $f \in X^\ast$.
From N... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $S \subseteq X$.
Suppos... | Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the [[Definition:Second Normed Dual|second normed dual]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\iota : X \to X^{\ast \ast}$ be the [[Definition:Evaluation Linear Transformation|evaluation linear transformation]].
We then have:
:$\ds \sup_... | Subset of Normed Vector Space providing Bounded Point Evaluations is Norm Bounded | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_providing_Bounded_Point_Evaluations_is_Norm_Bounded | https://proofwiki.org/wiki/Subset_of_Normed_Vector_Space_providing_Bounded_Point_Evaluations_is_Norm_Bounded | [
"Normed Dual Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Bounded Subset of Normed Vector Space"
] | [
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation",
"Normed Dual Space is Banach Space",
"Definition:Banach Space",
"Banach-Steinhaus Theorem",
"Evaluation Linear Transformation on Normed Vector Space is Linear Isometry",
"Definition:Linear Isometry",
"Category:Normed Dual ... |
proofwiki-22755 | Weakly Countably Compact Set in Normed Vector Space is Norm Bounded | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $C \subseteq X$ be $w$-countably compact.
Then:
:$\ds \sup_{x \mathop \in S} \norm x_X < \infty$
Hence $C$ is bounded. | Let $X^\ast$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $f \in X^\ast$.
From Continuous Image of Countably Compact Space is Countably Compact, $f \sqbrk C$ is countably compact in $\GF$.
From Countably Compact Metric Space is Bounded, $f \sqbrk C$ is bounded.
Hence for each $f \in X^\ast$ we ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $C \subseteq X$ be [[Definition:Countably Co... | Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $f \in X^\ast$.
From [[Continuous Image of Countably Compact Space is Countably Compact]], $f \sqbrk C$ is [[Definition:Countably Compact Space|countably compact]] in $\GF$.
From [[Countably Compact Me... | Weakly Countably Compact Set in Normed Vector Space is Norm Bounded | https://proofwiki.org/wiki/Weakly_Countably_Compact_Set_in_Normed_Vector_Space_is_Norm_Bounded | https://proofwiki.org/wiki/Weakly_Countably_Compact_Set_in_Normed_Vector_Space_is_Norm_Bounded | [
"Countably Compact Spaces",
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Countably Compact Space",
"Definition:Bounded Subset of Normed Vector Space"
] | [
"Definition:Normed Dual Space",
"Continuous Image of Countably Compact Space is Countably Compact",
"Definition:Countably Compact Space",
"Countably Compact Metric Space is Bounded",
"Definition:Bounded Subset of Normed Vector Space",
"Subset of Normed Vector Space providing Bounded Point Evaluations is N... |
proofwiki-22756 | Accumulation Point of Sequence iff Cluster Point of Net | Let $\struct {X, \tau}$ be a topological space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.
Let $x \in X$.
Then $x$ is an accumulation point of $\sequence {x_n}_{n \mathop \in \N}$ as a sequence {{iff}} $x$ is a cluster point of $\sequence {x_n}_{n \mathop \in \N}$ as a net. | === Necessary Condition ===
Suppose that $x$ is an accumulation point of $\sequence {x_n}_{n \mathop \in \N}$.
Let $U \in \tau$.
Then:
:$\SS = \set {n \in \N : x_n \in U}$ is infinite.
Note that $\set {1, \ldots, K}$ is finite and every subset of a finite set is finite by Subset of Finite Set is Finite.
Hence for each ... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$.
Let $x \in X$.
Then $x$ is an [[Definition:Accumulation Point/Sequence|accumulation point]] of $\sequence {x_n}_{n \mathop \in \N}$ as a [[Definitio... | === Necessary Condition ===
Suppose that $x$ is an [[Definition:Accumulation Point/Sequence|accumulation point]] of $\sequence {x_n}_{n \mathop \in \N}$.
Let $U \in \tau$.
Then:
:$\SS = \set {n \in \N : x_n \in U}$ is [[Definition:Infinite Set|infinite]].
Note that $\set {1, \ldots, K}$ is [[Definition:Finite Set|f... | Accumulation Point of Sequence iff Cluster Point of Net | https://proofwiki.org/wiki/Accumulation_Point_of_Sequence_iff_Cluster_Point_of_Net | https://proofwiki.org/wiki/Accumulation_Point_of_Sequence_iff_Cluster_Point_of_Net | [
"Nets (Set Theory)"
] | [
"Definition:Topological Space",
"Definition:Sequence",
"Definition:Accumulation Point/Sequence",
"Definition:Sequence",
"Definition:Convergent Net/Cluster Point",
"Definition:Net (Set Theory)"
] | [
"Definition:Accumulation Point/Sequence",
"Definition:Infinite Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Finite Set",
"Definition:Finite Set",
"Subset of Finite Set is Finite",
"Definition:Convergent Net/Cluster Point",
"Definition:Net (Set Theory)",
"Definition:Convergent Ne... |
proofwiki-22757 | Interior of Closed Rectangle | Let:
:$\ds \RR = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$
be a closed $n$-rectangle.
Then:
:$\ds \RR^\circ = \prod_{i \mathop = 1}^n \openint {a_i} {b_i}$
where $\RR^\circ$ denotes the interior of $\RR$. | {{begin-eqn}}
{{eqn | l = \RR^\circ
| r = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}^\circ
| c = Interior of Cartesian Product is Product of Interiors
}}
{{eqn | r = \prod_{i \mathop = 1}^n \openint {a_i} {b_i}
| c = Interior of Closed Real Interval is Open Real Interval
}}
{{end-eqn}}{{qed}}
Cate... | Let:
:$\ds \RR = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$
be a [[Definition:Closed Rectangle|closed $n$-rectangle]].
Then:
:$\ds \RR^\circ = \prod_{i \mathop = 1}^n \openint {a_i} {b_i}$
where $\RR^\circ$ denotes the [[Definition:Interior (Topology)|interior]] of $\RR$. | {{begin-eqn}}
{{eqn | l = \RR^\circ
| r = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}^\circ
| c = [[Interior of Cartesian Product is Product of Interiors]]
}}
{{eqn | r = \prod_{i \mathop = 1}^n \openint {a_i} {b_i}
| c = [[Interior of Closed Real Interval is Open Real Interval]]
}}
{{end-eqn}}{{qe... | Interior of Closed Rectangle | https://proofwiki.org/wiki/Interior_of_Closed_Rectangle | https://proofwiki.org/wiki/Interior_of_Closed_Rectangle | [
"Examples of Set Interiors"
] | [
"Definition:Closed Rectangle",
"Definition:Interior (Topology)"
] | [
"Interior of Cartesian Product is Product of Interiors",
"Interior of Closed Real Interval is Open Real Interval",
"Category:Examples of Set Interiors"
] |
proofwiki-22758 | Continuous Mapping Induced by Localic Mapping is Continuous | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be locales.
Let $f : L_1 \to L_2$ be a localic mapping.
Let:
:$\map {\operatorname{Sp}} f : \map {\operatorname{Sp}} {L_1} \to \map {\operatorname{Sp}} {L_2}$ denote the continuous mapping induced by $f$
where:
:$\map {\operatorname{Sp}} {L_1}$ and... | Let $\loweradjoint f : L_2 \to L_1$ denote the lower adjoint such that $\tuple{f, \loweradjoint f}$ is a Galois connection.
By definition of a localic mapping:
:$\loweradjoint f: L_2 \to L_1$ is a frame homomorphism.
Let:
:$\map {\operatorname{Sp}} {f^*} : \map {\operatorname{Sp}} {L_1} \to \map {\operatorname{Sp}} {L_... | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Locale (Lattice Theory)|locales]].
Let $f : L_1 \to L_2$ be a [[Definition:Localic Mapping|localic mapping]].
Let:
:$\map {\operatorname{Sp}} f : \map {\operatorname{Sp}} {L_1} \to \map {\operatorname{Sp}} {L_2}$ denote the [[Def... | Let $\loweradjoint f : L_2 \to L_1$ denote the [[Definition:Lower Adjoint|lower adjoint]] such that $\tuple{f, \loweradjoint f}$ is a [[Definition:Galois Connection|Galois connection]].
By definition of a [[Definition:Localic Mapping|localic mapping]]:
:$\loweradjoint f: L_2 \to L_1$ is a [[Definition:Frame Homomorph... | Continuous Mapping Induced by Localic Mapping is Continuous | https://proofwiki.org/wiki/Continuous_Mapping_Induced_by_Localic_Mapping_is_Continuous | https://proofwiki.org/wiki/Continuous_Mapping_Induced_by_Localic_Mapping_is_Continuous | [
"Continuous Maps",
"Continuous Mappings"
] | [
"Definition:Locale (Lattice Theory)",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Continuous Mapping Induced by Localic Mapping",
"Definition:Spectrum of Locale/Completely Prime Filters",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [
"Definition:Galois Connection/Lower Adjoint",
"Definition:Galois Connection",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Frame Homomorphism",
"Definition:Continuous Mapping Induced by Frame Homomorphism",
"Continuous Mapping Induced by Frame Homomorphism is Continuous",
"Definitio... |
proofwiki-22759 | Localic Spectrum Functor is Covariant Functor | Let $\mathbf{Loc_*}$ denote the category of locales with localic mappings.
Let $\mathbf{Top}$ denote the category of topological spaces.
Then:
:the localic spectrum functor $\operatorname {Sp} : \mathbf{Loc_*} \to \mathbf{Top}$ is a covariant functor | Let $\mathbf{Loc}$ denote the category of locales.
Let $G : \mathbf{Loc_*} \to \mathbf{Loc}$ be defined by:
:for each locale $L$ of $\mathbf{Loc_*} : \map G L = L$
:for each localic mapping $g : L_1 \to L_2$ of $\mathbf{Loc_*} : \map G g = \paren{g^*}^{\operatorname{op}}$
where:
:$g^* : L_2 \to L_1$ denotes the frame h... | Let $\mathbf{Loc_*}$ denote the [[Definition:Category of Locales with Localic Mappings|category of locales with localic mappings]].
Let $\mathbf{Top}$ denote the [[Definition:Category of Topological Spaces|category of topological spaces]].
Then:
:the [[Definition:Localic Spectrum Functor|localic spectrum functor]] $... | Let $\mathbf{Loc}$ denote the [[Definition:Category of Locales|category of locales]].
Let $G : \mathbf{Loc_*} \to \mathbf{Loc}$ be defined by:
:for each [[Definition:Locale (Lattice Theory)|locale]] $L$ of $\mathbf{Loc_*} : \map G L = L$
:for each [[Definition:Localic Mapping|localic mapping]] $g : L_1 \to L_2$ of $\... | Localic Spectrum Functor is Covariant Functor | https://proofwiki.org/wiki/Localic_Spectrum_Functor_is_Covariant_Functor | https://proofwiki.org/wiki/Localic_Spectrum_Functor_is_Covariant_Functor | [
"Functors"
] | [
"Definition:Category of Locales with Localic Mappings",
"Definition:Category of Topological Spaces",
"Definition:Localic Spectrum Functor",
"Definition:Functor/Covariant"
] | [
"Definition:Category of Locales",
"Definition:Locale (Lattice Theory)",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Frame Homomorphism",
"Definition:Galois Connection/Lower Adjoint",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Dual Category",
"Definition:Fra... |
proofwiki-22760 | Point in Closure of Sequence in Metric Space is Term of Sequence or Subsequential Limit | Let $\struct {X, d}$ be a metric space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.
Let $\cl$ be the closure taken in $X$.
Let $x \in \map \cl {\sequence {x_n}_{n \mathop \in \N} }$.
Then either there exists $n \in \N$ with $x = x_n$ or there exists a subsequence $\sequence {x_{n_k} }_{k \mathop \in ... | Suppose that $x_n \ne x$ for all $n \in \N$.
From Point in Closure of Subset of Metric Space iff Limit of Sequence, there exists a sequence $\sequence {y_n}_{n \mathop \in \N}$ in $\map \cl {\sequence {x_n}_{n \mathop \in \N} }$ such that $y_n \to x$ in $\struct {X, d}$.
Write $y_n = x_{\map \phi n}$.
Since $x_n \ne ... | Let $\struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$.
Let $\cl$ be the [[Definition:Topological Closure|closure]] taken in $X$.
Let $x \in \map \cl {\sequence {x_n}_{n \mathop \in \N} }$.
Then either there exists $... | Suppose that $x_n \ne x$ for all $n \in \N$.
From [[Point in Closure of Subset of Metric Space iff Limit of Sequence]], there exists a [[Definition:Sequence|sequence]] $\sequence {y_n}_{n \mathop \in \N}$ in $\map \cl {\sequence {x_n}_{n \mathop \in \N} }$ such that $y_n \to x$ in $\struct {X, d}$.
Write $y_n = x_{\... | Point in Closure of Sequence in Metric Space is Term of Sequence or Subsequential Limit | https://proofwiki.org/wiki/Point_in_Closure_of_Sequence_in_Metric_Space_is_Term_of_Sequence_or_Subsequential_Limit | https://proofwiki.org/wiki/Point_in_Closure_of_Sequence_in_Metric_Space_is_Term_of_Sequence_or_Subsequential_Limit | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Closure (Topology)",
"Definition:Subsequence",
"Definition:Convergent Sequence"
] | [
"Point in Closure of Subset of Metric Space iff Limit of Sequence",
"Definition:Sequence",
"Definition:Eventually Constant Sequence",
"Definition:Bounded Sequence/Unbounded",
"Definition:Subsequence",
"Definition:Subsequence",
"Limit of Subsequence equals Limit of Sequence",
"Definition:Convergent Seq... |
proofwiki-22761 | Characterization of Unconditional Convergence of Series in Banach Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.
{{TFAE}}
:$(1): \quad$ $\ds \sum_{n \mathop = 1}^\infty x_n$ converges unconditionally
:$(2): \quad$ for each strictly increasing sequence $\sequence {n_k}_{... | === $(4)$ implies $(1)$ ===
Suppose that:
:for each $\epsilon > 0$, there exists $n \in \N$ such that for all finite sets $F \subseteq \set {j : j \ge n + 1}$ we have $\ds \norm {\sum_{j \mathop \in F} x_j}_X < \epsilon$
Let $\pi : \N \to \N$ be a permutation.
Let $\epsilon > 0$.
Take $M \in \N$ such that for every fin... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$.
{{TFAE}}
:$(1): \quad$ $\ds \sum_{n \mathop = 1}^\infty x_n$ [[Definition:Unconditionally Convergent ... | === $(4)$ implies $(1)$ ===
Suppose that:
:for each $\epsilon > 0$, there exists $n \in \N$ such that for all [[Definition:Finite Set|finite sets]] $F \subseteq \set {j : j \ge n + 1}$ we have $\ds \norm {\sum_{j \mathop \in F} x_j}_X < \epsilon$
Let $\pi : \N \to \N$ be a [[Definition:Permutation|permutation]].
Let... | Characterization of Unconditional Convergence of Series in Banach Space | https://proofwiki.org/wiki/Characterization_of_Unconditional_Convergence_of_Series_in_Banach_Space | https://proofwiki.org/wiki/Characterization_of_Unconditional_Convergence_of_Series_in_Banach_Space | [
"Unconditionally Convergent Series"
] | [
"Definition:Banach Space",
"Definition:Sequence",
"Definition:Unconditionally Convergent Series",
"Definition:Strictly Increasing/Sequence",
"Definition:Convergent Series/Normed Vector Space",
"Definition:Convergent Series/Normed Vector Space",
"Definition:Finite Set"
] | [
"Definition:Finite Set",
"Definition:Permutation",
"Definition:Finite Set",
"Definition:Injection",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Sequence",
"Definition:Cauchy Sequence",
"Definition:Banach Space",
"Definition:Convergent Series/Normed Vector Space",
"Definition:Permut... |
proofwiki-22762 | Banach Space with Weak Topology has Countable Tightness | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Then $\struct {X, w}$ has countable tightness. | Let $A \subseteq X$.
Let $\cl_w$ be the closure taken in $\struct {X, w}$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm {\, \... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$.
Then $\struct {X, w}$ has [[Definition:Countable Tightness|co... | Let $A \subseteq X$.
Let $\cl_w$ be the [[Definition:Topological Closure|closure]] taken in $\struct {X, w}$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X... | Banach Space with Weak Topology has Countable Tightness | https://proofwiki.org/wiki/Banach_Space_with_Weak_Topology_has_Countable_Tightness | https://proofwiki.org/wiki/Banach_Space_with_Weak_Topology_has_Countable_Tightness | [
"Countable Tightness",
"Weak Topologies on Topological Vector Spaces",
"Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Countable Tightness"
] | [
"Definition:Closure (Topology)",
"Definition:Normed Dual Space",
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation",
"Definition:Closed Unit Ball",
"Definition:Weak-* Topology",
"Definition:Set",
"Definition:Ordered Tuple",
"Open Sets in Weak Topology of Topological Vecto... |
proofwiki-22763 | Weakly Compact Set in Banach Space is Angelic | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\struct {K, w}$ be compact.
Then $\struct {K, w}$ is angelic. | Let $\cl_w$ be the closure taken in $\struct {X, w}$.
Let $\cl$ be the closure taken in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $A \subseteq K$.
Let $x \in \map {\cl_w} A$.
From Banach Space with Weak Topology has Countable Tightness, $\struct {X, w}$ has countable tightness.
Hence there exists a countable set $S \s... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\struct {K, w}$ be [[Definition:Compact Topological Space... | Let $\cl_w$ be the [[Definition:Topological Closure|closure]] taken in $\struct {X, w}$.
Let $\cl$ be the [[Definition:Topological Closure|closure]] taken in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $A \subseteq K$.
Let $x \in \map {\cl_w} A$.
From [[Banach Space with Weak Topology has Countable Tightness]], $\st... | Weakly Compact Set in Banach Space is Angelic | https://proofwiki.org/wiki/Weakly_Compact_Set_in_Banach_Space_is_Angelic | https://proofwiki.org/wiki/Weakly_Compact_Set_in_Banach_Space_is_Angelic | [
"Weakly Compact Sets",
"Angelic Topological Spaces"
] | [
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Compact Topological Space",
"Definition:Angelic Topological Space"
] | [
"Definition:Closure (Topology)",
"Definition:Closure (Topology)",
"Banach Space with Weak Topology has Countable Tightness",
"Definition:Countable Tightness",
"Definition:Countable Set",
"Characterization of Separable Normed Vector Space",
"Definition:Separable Space",
"Topological Closure is Closed",... |
proofwiki-22764 | Weakly Compact Set in Banach Space whose Normed Dual is Weak-* Separable is Weakly Metrizable | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space.
Let $w$ be the weak topology on $X$.
Let $\struct {K, w}$ be compact with $K \subseteq X$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w^\ast$ be the ... | Let $\SS_\ast$ be a countable everywhere dense subset of $\struct {X^\ast, w^\ast}$.
From Weak-* Dense Subset of Normed Dual Space Separates Points, $\SS_\ast$ separates points.
Let $\SS = \set {f \restriction_K : f \in \SS_\ast}$.
Let $\tau$ be the initial topology on $K$ generated by $\SS$.
From Initial Topology Gen... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]].
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $X$.
Let $\struct {K, w}$ be [[Definition:Compact Topological Space|compact]] with $K \subseteq X$.
Let $\stru... | Let $\SS_\ast$ be a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere dense subset]] of $\struct {X^\ast, w^\ast}$.
From [[Weak-* Dense Subset of Normed Dual Space Separates Points]], $\SS_\ast$ [[Definition:Mappings Separating Points|separates points]].
Let $\SS = \set {f \restriction_K... | Weakly Compact Set in Banach Space whose Normed Dual is Weak-* Separable is Weakly Metrizable | https://proofwiki.org/wiki/Weakly_Compact_Set_in_Banach_Space_whose_Normed_Dual_is_Weak-*_Separable_is_Weakly_Metrizable | https://proofwiki.org/wiki/Weakly_Compact_Set_in_Banach_Space_whose_Normed_Dual_is_Weak-*_Separable_is_Weakly_Metrizable | [
"Weakly Compact Sets",
"Weak-* Topologies",
"Metrizable Spaces"
] | [
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Compact Topological Space",
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Definition:Separable Space",
"Definition:Metrizable Space"
] | [
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Weak-* Dense Subset of Normed Dual Space Separates Points",
"Definition:Mappings Separating Points",
"Definition:Initial Topology",
"Initial Topology Generated by Countable Family of Functions Separating Points is Metrizable",
"Definition:Metri... |
proofwiki-22765 | Weakly Compact Set in Separable Banach Space is Weakly Metrizable | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a separable Banach space.
Let $w$ be the weak topology on $X$.
Let $\struct {K, w}$ be compact with $K \subseteq X$.
Let $\struct {K, w}$ is metrizable. | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.
From Normed Dual Space of Separable Normed Vector Space is Weak-* Separable, $\struct {X^\ast, w^\ast}$ is separable.
From Weakly Compact Set in B... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Separable Space|separable]] [[Definition:Banach Space|Banach space]].
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $X$.
Let $\struct {K, w}$ be [[Definition:Compact Topological Space|co... | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w^\ast$ be the [[Definition:Weak-* Topology|weak-$\ast$ topology]] on $X^\ast$.
From [[Normed Dual Space of Separable Normed Vector Space is Weak-* Separable]],... | Weakly Compact Set in Separable Banach Space is Weakly Metrizable | https://proofwiki.org/wiki/Weakly_Compact_Set_in_Separable_Banach_Space_is_Weakly_Metrizable | https://proofwiki.org/wiki/Weakly_Compact_Set_in_Separable_Banach_Space_is_Weakly_Metrizable | [
"Weakly Compact Sets",
"Metrizable Spaces",
"Separable Spaces",
"Banach Spaces"
] | [
"Definition:Separable Space",
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Compact Topological Space",
"Definition:Metrizable Space"
] | [
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Normed Dual Space of Separable Normed Vector Space is Weak-* Separable",
"Definition:Separable Space",
"Weakly Compact Set in Banach Space whose Normed Dual is Weak-* Separable is Weakly Metrizable",
"Definition:Metrizable Space",
"Category:... |
proofwiki-22766 | Closed Vector Subspace in Normed Vector Space is Weakly Closed | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $Y$ be a closed vector subspace of $X$.
Then $Y$ is weakly closed. | From Linear Subspace is Convex Set, $Y$ is convex.
Further, $Y$ is norm closed in $\struct {X, \norm {\, \cdot \,}_X}$.
From Mazur's Theorem: Corollary, $Y$ is weakly closed.
{{qed}}
Category:Normed Vector Spaces
d800cw8h3af1ldz56zg7syvq6rfhglr | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $Y$ be a [[Definition:Closed Linear Subspace|closed vector subspace]] of $X$.
Then $Y$ is [[Definition:Weakly Closed Set|weakly closed]]. | From [[Linear Subspace is Convex Set]], $Y$ is [[Definition:Convex Set|convex]].
Further, $Y$ is [[Definition:Closed Set|norm closed]] in $\struct {X, \norm {\, \cdot \,}_X}$.
From [[Mazur's Theorem/Corollary|Mazur's Theorem: Corollary]], $Y$ is [[Definition:Weakly Closed Set|weakly closed]].
{{qed}}
[[Category:Nor... | Closed Vector Subspace in Normed Vector Space is Weakly Closed | https://proofwiki.org/wiki/Closed_Vector_Subspace_in_Normed_Vector_Space_is_Weakly_Closed | https://proofwiki.org/wiki/Closed_Vector_Subspace_in_Normed_Vector_Space_is_Weakly_Closed | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Linear Subspace",
"Definition:Weakly Closed Set"
] | [
"Linear Subspace is Convex Set",
"Definition:Convex Set",
"Definition:Closed Set",
"Mazur's Theorem/Corollary",
"Definition:Weakly Closed Set",
"Category:Normed Vector Spaces"
] |
proofwiki-22767 | Bounded Subset of Banach Space is Relatively Weakly Compact iff Weak-* Closure in Second Normed Dual is Contained in Embedding of Original Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $S \subseteq X$ be bounded.
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w^\ast$ be the weak-$\ast$ topology on $\struct... | === Necessary Condition ===
Suppose that $\struct {\map {\cl_w} S, w}$ is compact.
Let $\phi \in \map {\cl_{w^\ast} } {\iota S}$.
Then by Point in Set Closure iff Limit of Net, there exists a directed set $\struct {\Lambda, \preceq}$ and a net $\family {\phi_\lambda}_{\lambda \mathop \in \Lambda}$ in $\iota S$ convergi... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $S \subseteq X$ be [[Definition:Bounded Subset of Normed Vector Space|bounded]].
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the [[Definition:Second Normed D... | === Necessary Condition ===
Suppose that $\struct {\map {\cl_w} S, w}$ is [[Definition:Compact Topological Space|compact]].
Let $\phi \in \map {\cl_{w^\ast} } {\iota S}$.
Then by [[Point in Set Closure iff Limit of Net]], there exists a [[Definition:Directed Set|directed set]] $\struct {\Lambda, \preceq}$ and a [[De... | Bounded Subset of Banach Space is Relatively Weakly Compact iff Weak-* Closure in Second Normed Dual is Contained in Embedding of Original Space | https://proofwiki.org/wiki/Bounded_Subset_of_Banach_Space_is_Relatively_Weakly_Compact_iff_Weak-*_Closure_in_Second_Normed_Dual_is_Contained_in_Embedding_of_Original_Space | https://proofwiki.org/wiki/Bounded_Subset_of_Banach_Space_is_Relatively_Weakly_Compact_iff_Weak-*_Closure_in_Second_Normed_Dual_is_Contained_in_Embedding_of_Original_Space | [
"Weakly Compact Sets",
"Weak-* Topologies"
] | [
"Definition:Banach Space",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Second Normed Dual",
"Definition:Weak-* Topology",
"Definition:Closure (Topology)",
"Definition:Closure (Topology)",
"Definition:Evaluation Linear Transformation",
"Definition:Relatively Compact Subspace",
"De... | [
"Definition:Compact Topological Space",
"Point in Set Closure iff Limit of Net",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Weak-* Topology",
"Definition:Net (Set Theory)",
"Definition:Compact Topological Space",
"Definition:Directed Pr... |
proofwiki-22768 | Bounded Subset of Normed Dual Space is Relatively Weak-* Compact | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.
Let $\cl_{w^\ast}$ be the closure taken in $\... | Let $B_{X^\ast}^-$ be the closed unit ball in $X^\ast$.
Let:
:$\ds \delta = \sup_{f \mathop \in S} \norm f_{X^\ast}$
Then we have:
:$S \subseteq \delta B_{X^\ast}^-$
From the Banach-Alaoglu Theorem, $\struct {B_{X^\ast}^-, w^\ast}$ is compact.
Hence from Dilation of Compact Set in Topological Vector Space is Compact, ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w^\ast$ be the [[Definition:Wea... | Let $B_{X^\ast}^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $X^\ast$.
Let:
:$\ds \delta = \sup_{f \mathop \in S} \norm f_{X^\ast}$
Then we have:
:$S \subseteq \delta B_{X^\ast}^-$
From the [[Banach-Alaoglu Theorem]], $\struct {B_{X^\ast}^-, w^\ast}$ is [[Definition:Compact Topological Space|compac... | Bounded Subset of Normed Dual Space is Relatively Weak-* Compact | https://proofwiki.org/wiki/Bounded_Subset_of_Normed_Dual_Space_is_Relatively_Weak-*_Compact | https://proofwiki.org/wiki/Bounded_Subset_of_Normed_Dual_Space_is_Relatively_Weak-*_Compact | [
"Weak-* Topologies"
] | [
"Definition:Banach Space",
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Definition:Closure (Topology)",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Relatively Compact Subspace",
"Definition:Compact Topological Space"
] | [
"Definition:Closed Unit Ball",
"Banach-Alaoglu Theorem",
"Definition:Compact Topological Space",
"Dilation of Compact Set in Topological Vector Space is Compact",
"Definition:Compact Topological Space",
"Weak-* Topology is Hausdorff",
"Definition:T2 Space",
"Compact Subspace of Hausdorff Space is Clos... |
proofwiki-22769 | Tail of Directed Set is Directed | Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\lambda_0 \in \Lambda$.
Let $\Lambda_0 = \set {\lambda \in \Lambda : \lambda \succeq \lambda_0}$.
Let $\preceq_0$ be the restriction of $\preceq$ to $\Lambda_0$.
Then $\struct {\Lambda_0, \preceq_0}$ is directed. | Let $\mu_1, \mu_2 \in \Lambda_0$.
We can find $\lambda \in \Lambda$ such that $\mu_1 \preceq \lambda$ and $\mu_2 \preceq \lambda$.
Since $\lambda_0 \preceq \mu_1$ and $\lambda_0 \preceq \mu_2$, we have that $\lambda_0 \preceq \lambda$ by transitivity.
Hence $\lambda \in \Lambda_0$, with $\mu_1 \preceq \lambda$ and $\m... | Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\lambda_0 \in \Lambda$.
Let $\Lambda_0 = \set {\lambda \in \Lambda : \lambda \succeq \lambda_0}$.
Let $\preceq_0$ be the [[Definition:Restriction of Relation|restriction]] of $\preceq$ to $\Lambda_0$.
Then $\struct {\Lambda_0, \pr... | Let $\mu_1, \mu_2 \in \Lambda_0$.
We can find $\lambda \in \Lambda$ such that $\mu_1 \preceq \lambda$ and $\mu_2 \preceq \lambda$.
Since $\lambda_0 \preceq \mu_1$ and $\lambda_0 \preceq \mu_2$, we have that $\lambda_0 \preceq \lambda$ by [[Definition:Transitive Relation|transitivity]].
Hence $\lambda \in \Lambda_0$... | Tail of Directed Set is Directed | https://proofwiki.org/wiki/Tail_of_Directed_Set_is_Directed | https://proofwiki.org/wiki/Tail_of_Directed_Set_is_Directed | [
"Directed Sets",
"Directed Preorderings",
"Directed Preorderings"
] | [
"Definition:Directed Preordering",
"Definition:Restriction/Relation",
"Definition:Directed Preordering"
] | [
"Definition:Transitive Relation",
"Definition:Directed Preordering",
"Category:Directed Preorderings"
] |
proofwiki-22770 | Translation of Bounded Subset of Normed Vector Space is Bounded | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $A \subseteq X$ be a bounded set.
Let $x \in X$.
Then $A + x$ is bounded. | Since $A$ is bounded, there exists $M > 0$ such that:
:$\norm y_X \le M$ for all $y \in A$.
Hence from {{NormAxiomVector|3}}, we have:
:$\norm {y + x}_X \le M + \norm x_X < \infty$ for all $y \in A$.
Hence $\norm u_X \le M + \norm x_X$ for all $u \in A + x$.
Hence $A + x$ is bounded.
{{qed}}
Category:Normed Vector Spac... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $A \subseteq X$ be a [[Definition:Bounded Subset of Normed Vector Space|bounded set]].
Let $x \in X$.
Then $A + x$ is [[Definition:Bounded Subset of Normed Vector Space|bounded]]. | Since $A$ is [[Definition:Bounded Subset of Normed Vector Space|bounded]], there exists $M > 0$ such that:
:$\norm y_X \le M$ for all $y \in A$.
Hence from {{NormAxiomVector|3}}, we have:
:$\norm {y + x}_X \le M + \norm x_X < \infty$ for all $y \in A$.
Hence $\norm u_X \le M + \norm x_X$ for all $u \in A + x$.
Hence... | Translation of Bounded Subset of Normed Vector Space is Bounded | https://proofwiki.org/wiki/Translation_of_Bounded_Subset_of_Normed_Vector_Space_is_Bounded | https://proofwiki.org/wiki/Translation_of_Bounded_Subset_of_Normed_Vector_Space_is_Bounded | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Bounded Subset of Normed Vector Space"
] | [
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Bounded Subset of Normed Vector Space",
"Category:Normed Vector Spaces"
] |
proofwiki-22771 | Topological Closure of Translation in Topological Vector Space is Translation of Topological Closure | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $x \in X$.
Let $A \subseteq X$ be non-empty.
Let $\cl$ be closure taken in $\struct {X, \tau}$.
Then $\map \cl {A + x} = \map \cl A + x$. | From Characterization of Closedness in terms of Nets, we have $y \in \map \cl {A + x}$ {{iff}} there exists a directed set $\struct {\Lambda, \preceq}$ and a net $\family {u_\lambda + x}_{\lambda \mathop \in \Lambda}$ with $u_\lambda \in A$ for each $\lambda \in \Lambda$ such that:
:$\family {u_\lambda + x}_{\lambda \m... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $x \in X$.
Let $A \subseteq X$ be [[Definition:Non-Empty Set|non-empty]].
Let $\cl$ be [[Definition:Topological Closure|closure]] taken in $\struct {X, \tau}$.
Then $\map \cl... | From [[Characterization of Closedness in terms of Nets]], we have $y \in \map \cl {A + x}$ {{iff}} there exists a [[Definition:Directed Set|directed set]] $\struct {\Lambda, \preceq}$ and a [[Definition:Net (Set Theory)|net]] $\family {u_\lambda + x}_{\lambda \mathop \in \Lambda}$ with $u_\lambda \in A$ for each $\lamb... | Topological Closure of Translation in Topological Vector Space is Translation of Topological Closure | https://proofwiki.org/wiki/Topological_Closure_of_Translation_in_Topological_Vector_Space_is_Translation_of_Topological_Closure | https://proofwiki.org/wiki/Topological_Closure_of_Translation_in_Topological_Vector_Space_is_Translation_of_Topological_Closure | [
"Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Non-Empty Set",
"Definition:Closure (Topology)"
] | [
"Characterization of Closedness in terms of Nets",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Constant Sequence in Topological Space Converges",
"Linear Combination of Convergent Sequences in Topological Vector Space is Convergent",
"Definition:Conver... |
proofwiki-22772 | Complex-Valued Function is Continuous iff Real Part and Imaginary Part are Continuous | Let $\struct {X, \tau}$ be a topological space.
Let $f : X \to \C$ be a function.
Then $f$ is continuous {{iff}} both the real part $\map \Re f$ and the imaginary part $\map \Im f$ are continuous. | === Necessary Condition ===
Suppose that $f$ is continuous.
From Real and Imaginary Part Projections are Continuous, $z \mapsto \map \Re z$ and $z \mapsto \map \Im z$ are continuous.
From Composite of Continuous Mappings is Continuous, $\map \Re f$ and $\map \Im f$ are continuous.
{{qed|lemma}} | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f : X \to \C$ be a [[Definition:Function|function]].
Then $f$ is [[Definition:Continuous Mapping|continuous]] {{iff}} both the [[Definition:Real Part|real part]] $\map \Re f$ and the [[Definition:Imaginary Part|imaginary part]] $\m... | === Necessary Condition ===
Suppose that $f$ is [[Definition:Continuous Mapping|continuous]].
From [[Real and Imaginary Part Projections are Continuous]], $z \mapsto \map \Re z$ and $z \mapsto \map \Im z$ are [[Definition:Continuous Mapping|continuous]].
From [[Composite of Continuous Mappings is Continuous]], $\map... | Complex-Valued Function is Continuous iff Real Part and Imaginary Part are Continuous | https://proofwiki.org/wiki/Complex-Valued_Function_is_Continuous_iff_Real_Part_and_Imaginary_Part_are_Continuous | https://proofwiki.org/wiki/Complex-Valued_Function_is_Continuous_iff_Real_Part_and_Imaginary_Part_are_Continuous | [
"Complex-Valued Functions"
] | [
"Definition:Topological Space",
"Definition:Function",
"Definition:Continuous Mapping",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part",
"Definition:Continuous Mapping"
] | [
"Definition:Continuous Mapping",
"Real and Imaginary Part Projections are Continuous",
"Definition:Continuous Mapping",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Mapping",
"Definition:Continuous Mapping",
"Definition:Continuous Mapping"
] |
proofwiki-22773 | Eberlein-Šmulian Characterization of Reflexive Banach Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Then $\struct {X, \norm {\, \cdot \,}_X}$ is reflexive {{iff}}:
:every bounded sequence $\sequence {x_n}_{n \mathop \in \N}$ has a weakly convergent subsequence. | Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$. | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Then $\struct {X, \norm {\, \cdot \,}_X}$ is [[Definition:Reflexive Space|reflexive]] {{iff}}:
:every [[Definition:Bounded Sequence|bounded sequence]] $\sequence {x_n}_{n \mathop \in \N}$ h... | Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$. | Eberlein-Šmulian Characterization of Reflexive Banach Space | https://proofwiki.org/wiki/Eberlein-Šmulian_Characterization_of_Reflexive_Banach_Space | https://proofwiki.org/wiki/Eberlein-Šmulian_Characterization_of_Reflexive_Banach_Space | [
"Eberlein-Šmulian Characterization of Reflexive Banach Space",
"Reflexive Spaces",
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Banach Space",
"Definition:Reflexive Space",
"Definition:Bounded Sequence",
"Definition:Weak Convergence",
"Definition:Subsequence"
] | [
"Definition:Weak Topology on Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space"
] |
proofwiki-22774 | 1-Sequence Space Continuously Surjects into Separable Banach Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$ be the $1$-sequence space.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a separable Banach space over $\GF$.
Then there exists a bounded linear surjection $Q : \map {\ell_1} \GF \to X$. | Let $B_X$ be the open unit ball in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_X^-$ be the closed unit ball in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_{\ell_1}$ be the open unit ball in $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$.
Let $B_{\ell_1}^-$ be the closed unit ball in $\struct {\map {\ell_1} \G... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$ be the [[Definition:p-Sequence Space|$1$-sequence space]].
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Separable Space|separable]] [[Definition:Banach Space|Banach space]] over $\GF$.
Then there exists a [[Definitio... | Let $B_X$ be the [[Definition:Open Unit Ball|open unit ball]] in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_X^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_{\ell_1}$ be the [[Definition:Open Unit Ball|open unit ball]] in $\struct {\map {\ell_1} \GF, \nor... | 1-Sequence Space Continuously Surjects into Separable Banach Space | https://proofwiki.org/wiki/1-Sequence_Space_Continuously_Surjects_into_Separable_Banach_Space | https://proofwiki.org/wiki/1-Sequence_Space_Continuously_Surjects_into_Separable_Banach_Space | [
"Banach Spaces",
"Separable Spaces",
"p-Sequence Spaces"
] | [
"Definition:p-Sequence Space",
"Definition:Separable Space",
"Definition:Banach Space",
"Definition:Bounded Linear Transformation",
"Definition:Surjection"
] | [
"Definition:Open Unit Ball",
"Definition:Closed Unit Ball",
"Definition:Open Unit Ball",
"Definition:Closed Unit Ball",
"Subspace of Separable Metric Space is Separable",
"Definition:Separable Space",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Absolutely Convergent Series in Normed ... |
proofwiki-22775 | Separable Banach Space is Linearly Isomorphic to Quotient of 1-Sequence Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$ be the $1$-sequence space.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a separable Banach space over $\GF$.
Then there exists a closed vector subspace $M$ of $\map {\ell_1} \GF$ such that:
:$X$ is linearly isomorphic to the normed quo... | From 1-Sequence Space Continuously Surjects into Separable Banach Space, there exists a bounded linear surjection $Q : \map {\ell_1} \GF \to X$.
From Kernel of Bounded Linear Transformation is Closed Linear Subspace, $\ker Q$ is a closed vector subspace of $X$.
Let $M = \ker Q$.
Then the normed quotient vector space $\... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$ be the [[Definition:p-Sequence Space|$1$-sequence space]].
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Separable Space|separable]] [[Definition:Banach Space|Banach space]] over $\GF$.
Then there exists a [[Definitio... | From [[1-Sequence Space Continuously Surjects into Separable Banach Space]], there exists a [[Definition:Bounded Linear Transformation|bounded linear]] [[Definition:Surjection|surjection]] $Q : \map {\ell_1} \GF \to X$.
From [[Kernel of Bounded Linear Transformation is Closed Linear Subspace]], $\ker Q$ is a [[Definit... | Separable Banach Space is Linearly Isomorphic to Quotient of 1-Sequence Space | https://proofwiki.org/wiki/Separable_Banach_Space_is_Linearly_Isomorphic_to_Quotient_of_1-Sequence_Space | https://proofwiki.org/wiki/Separable_Banach_Space_is_Linearly_Isomorphic_to_Quotient_of_1-Sequence_Space | [
"Separable Spaces",
"p-Sequence Spaces",
"Quotient Normed Vector Spaces",
"Banach Spaces"
] | [
"Definition:p-Sequence Space",
"Definition:Separable Space",
"Definition:Banach Space",
"Definition:Closed Linear Subspace",
"Definition:Linear Isomorphism",
"Definition:Normed Quotient Vector Space"
] | [
"1-Sequence Space Continuously Surjects into Separable Banach Space",
"Definition:Bounded Linear Transformation",
"Definition:Surjection",
"Kernel of Bounded Linear Transformation is Closed Linear Subspace",
"Definition:Closed Linear Subspace",
"Definition:Normed Quotient Vector Space",
"Continuous Surj... |
proofwiki-22776 | Weak Compactness is Equivalent to Norm Compactness in Schur Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space that is a Schur space.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $K \subseteq X$.
Then $\struct {K, w}$ is compact {{iff}} $\struct {K, \norm {\, \cdot \,}_X}$ is compact. | === Necessary Condition ===
Suppose that $\struct {K, w}$ is compact.
From the Eberlein-Šmulian Theorem, $\struct {K, w}$ is sequentially compact.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $K$.
Since $\struct {K, w}$ is sequentially compact, there exists a subsequence $\sequence {x_{n_k} }_{k \mathop \i... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] that is a [[Definition:Schur Space|Schur space]].
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $K \subseteq X$.
T... | === Necessary Condition ===
Suppose that $\struct {K, w}$ is [[Definition:Compact Topological Space|compact]].
From the [[Eberlein-Šmulian Theorem]], $\struct {K, w}$ is [[Definition:Sequentially Compact Space|sequentially compact]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $... | Weak Compactness is Equivalent to Norm Compactness in Schur Space | https://proofwiki.org/wiki/Weak_Compactness_is_Equivalent_to_Norm_Compactness_in_Schur_Space | https://proofwiki.org/wiki/Weak_Compactness_is_Equivalent_to_Norm_Compactness_in_Schur_Space | [
"Weakly Compact Sets",
"Schur Spaces"
] | [
"Definition:Banach Space",
"Definition:Schur Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space"
] | [
"Definition:Compact Topological Space",
"Eberlein-Šmulian Theorem",
"Definition:Sequentially Compact Space",
"Definition:Sequence",
"Definition:Sequentially Compact Space",
"Definition:Subsequence",
"Definition:Weak Convergence",
"Definition:Schur Space",
"Definition:Convergent Sequence/Normed Vecto... |
proofwiki-22777 | Weak Topology and Norm Topology on Normed Vector Space Coincide iff Finite Dimensional | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $d$ be the metric induced on $X$ by $\norm {\, \cdot \,}$.
Let $\tau$ be the topology on $X$ induced by $d$.
Let $w$ be the weak topology on $X$.
Then $\tau = w$ {{iff}} $X$ is finite-dimensional. | === Necessary Condition ===
Suppose that $X$ is not finite-dimensional.
Since $\tau$ is induced by a metric, it is metrizable.
From Weak Topology on Infinite Dimensional Normed Vector Space is not Metrizable, $w$ is not metrizable.
Hence $\tau \ne w$.
{{qed|lemma}} | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $d$ be the [[Definition:Metric Induced by Norm|metric induced]] on $X$ by $\norm {\, \cdot \,}$.
Let $\tau$ be the [[Definition:Topology Induced by Metric|topology on $X$ ... | === Necessary Condition ===
Suppose that $X$ is not [[Definition:Finite-Dimensional Vector Space|finite-dimensional]].
Since $\tau$ is [[Definition:Topology Induced by Metric|induced by a metric]], it is [[Definition:Metrizable Space|metrizable]].
From [[Weak Topology on Infinite Dimensional Normed Vector Space is n... | Weak Topology and Norm Topology on Normed Vector Space Coincide iff Finite Dimensional | https://proofwiki.org/wiki/Weak_Topology_and_Norm_Topology_on_Normed_Vector_Space_Coincide_iff_Finite_Dimensional | https://proofwiki.org/wiki/Weak_Topology_and_Norm_Topology_on_Normed_Vector_Space_Coincide_iff_Finite_Dimensional | [
"Normed Vector Spaces",
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Metric Induced by Norm",
"Definition:Topology Induced by Metric",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Dimension of Vector Space/Finite"
] | [
"Definition:Dimension of Vector Space/Finite",
"Definition:Topology Induced by Metric",
"Definition:Metrizable Space",
"Weak Topology on Infinite Dimensional Normed Vector Space is not Metrizable",
"Definition:Metrizable Space",
"Definition:Dimension of Vector Space/Finite"
] |
proofwiki-22778 | Topologies with Same Convergent Nets are Equal | Let $X$ be a set.
Let $\tau_1$ and $\tau_2$ be topologies on $X$ such that:
:a net converges to $x$ in $\struct {X, \tau_1}$ {{iff}} it converges to $x$ in $\struct {X, \tau_2}$.
Then $\tau_1 = \tau_2$. | Let $U \subseteq X$.
By Characterization of Openness in terms of Nets, we have that $U \in \tau_1$ {{iff}} for each:
:$x \in U$
:directed set $\tuple {\Lambda, \preceq}$
:net $\family {x_\lambda}_{\lambda \in \Lambda}$ converging to $x$ in $\struct {X, \tau_1}$
there exists $\lambda \in \Lambda$ with $x_\lambda \in U$.... | Let $X$ be a [[Definition:Set|set]].
Let $\tau_1$ and $\tau_2$ be [[Definition:Topology|topologies]] on $X$ such that:
:a [[Definition:Net (Set Theory)|net]] [[Definition:Convergent Net|converges]] to $x$ in $\struct {X, \tau_1}$ {{iff}} it [[Definition:Convergent Net|converges]] to $x$ in $\struct {X, \tau_2}$.
The... | Let $U \subseteq X$.
By [[Characterization of Openness in terms of Nets]], we have that $U \in \tau_1$ {{iff}} for each:
:$x \in U$
:[[Definition:Directed Set|directed set]] $\tuple {\Lambda, \preceq}$
:[[Definition:Net (Set Theory)|net]] $\family {x_\lambda}_{\lambda \in \Lambda}$ [[Definition:Convergent Net|convergi... | Topologies with Same Convergent Nets are Equal | https://proofwiki.org/wiki/Topologies_with_Same_Convergent_Nets_are_Equal | https://proofwiki.org/wiki/Topologies_with_Same_Convergent_Nets_are_Equal | [
"Nets (Set Theory)"
] | [
"Definition:Set",
"Definition:Topology",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Convergent Net"
] | [
"Characterization of Openness in terms of Nets",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Characterization of Openness in terms of Nets",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:... |
proofwiki-22779 | Reflexive Space is Banach Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a reflexive normed vector space.
Then $\struct {X, \norm {\, \cdot \,}_X}$ is a Banach space. | Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\iota : X \to X^{\ast \ast}$ be the evaluation linear transformation.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {X, \norm {\, \cdot \,}_X}$.
From E... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Reflexive Space|reflexive]] [[Definition:Normed Vector Space|normed vector space]].
Then $\struct {X, \norm {\, \cdot \,}_X}$ is a [[Definition:Banach Space|Banach space]]. | Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the [[Definition:Second Normed Dual|second normed dual]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\iota : X \to X^{\ast \ast}$ be the [[Definition:Evaluation Linear Transformation|evaluation linear transformation]].
Let $\sequence {x_n}_{n \... | Reflexive Space is Banach Space | https://proofwiki.org/wiki/Reflexive_Space_is_Banach_Space | https://proofwiki.org/wiki/Reflexive_Space_is_Banach_Space | [
"Reflexive Spaces",
"Banach Spaces"
] | [
"Definition:Reflexive Space",
"Definition:Normed Vector Space",
"Definition:Banach Space"
] | [
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation",
"Definition:Cauchy Sequence",
"Evaluation Linear Transformation on Normed Vector Space is Linear Isometry",
"Definition:Cauchy Sequence",
"Normed Dual Space is Banach Space",
"Definition:Banach Space",
"Definition:Isometri... |
proofwiki-22780 | Reflexive Banach Space is Schur Space iff Finite Dimensional | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a reflexive Banach space over $\GF$.
Then $\struct {X, \norm {\, \cdot \,}_X}$ is a Schur space {{iff}} it is finite-dimensional. | === Necessary Condition ===
Suppose that $\struct {X, \norm {\, \cdot \,}_X}$ is a Schur space.
Let $B_X^-$ be the closed unit ball in $\struct {X, \norm {\, \cdot \,}_X}$.
Let:
:$S_X = \set {x \in X : \norm x_X = 1}$
Since $\struct {X, \norm {\, \cdot \,}_X}$ is reflexive, $B_X^-$ is weakly compact from Kakutani's The... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Reflexive Space|reflexive]] [[Definition:Banach Space|Banach space]] over $\GF$.
Then $\struct {X, \norm {\, \cdot \,}_X}$ is a [[Definition:Schur Space|Schur space]] {{iff}} it is [[Definition:Finite-Dimensional Vector Space|fin... | === Necessary Condition ===
Suppose that $\struct {X, \norm {\, \cdot \,}_X}$ is a [[Definition:Schur Space|Schur space]].
Let $B_X^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $\struct {X, \norm {\, \cdot \,}_X}$.
Let:
:$S_X = \set {x \in X : \norm x_X = 1}$
Since $\struct {X, \norm {\, \cdot \,}_... | Reflexive Banach Space is Schur Space iff Finite Dimensional | https://proofwiki.org/wiki/Reflexive_Banach_Space_is_Schur_Space_iff_Finite_Dimensional | https://proofwiki.org/wiki/Reflexive_Banach_Space_is_Schur_Space_iff_Finite_Dimensional | [
"Reflexive Spaces",
"Schur Spaces",
"Banach Spaces"
] | [
"Definition:Reflexive Space",
"Definition:Banach Space",
"Definition:Schur Space",
"Definition:Dimension of Vector Space/Finite"
] | [
"Definition:Schur Space",
"Definition:Closed Unit Ball",
"Definition:Reflexive Space",
"Definition:Weakly Compact Set",
"Kakutani's Theorem",
"Weak Compactness is Equivalent to Norm Compactness in Schur Space",
"Definition:Compact Topological Space",
"Closed Subspace of Compact Space is Compact",
"D... |
proofwiki-22781 | Subsequence Characterization of Cauchy Sequence in Topological Vector Space | Let $K$ be a topological field.
Let $\struct {X, \tau}$ be a topological vector space over $K$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.
Then $\sequence {x_n}_{n \mathop \in \N}$ is a Cauchy sequence {{iff}}:
:for all strictly increasing sequences $\sequence {n_k}_{k \mathop \in \N}$ and $\sequenc... | === Necessary Condition ===
Suppose that $\sequence {x_n}_{n \mathop \in \N}$ is a Cauchy sequence.
Let $\sequence {n_k}_{k \mathop \in \N}$ and $\sequence {m_k}_{k \mathop \in \N}$ in $\N$ be strictly increasing sequences.
Let $U$ be an open neighborhood of ${\mathbf 0}_X$ in $\struct {X, \tau}$.
There exists $N \in ... | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $K$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$.
Then $\sequence {x_n}_{n \mathop \in \N}$ is a [[Definition:... | === Necessary Condition ===
Suppose that $\sequence {x_n}_{n \mathop \in \N}$ is a [[Definition:Cauchy Sequence/Topological Vector Space|Cauchy sequence]].
Let $\sequence {n_k}_{k \mathop \in \N}$ and $\sequence {m_k}_{k \mathop \in \N}$ in $\N$ be [[Definition:Strictly Increasing Sequence|strictly increasing sequenc... | Subsequence Characterization of Cauchy Sequence in Topological Vector Space | https://proofwiki.org/wiki/Subsequence_Characterization_of_Cauchy_Sequence_in_Topological_Vector_Space | https://proofwiki.org/wiki/Subsequence_Characterization_of_Cauchy_Sequence_in_Topological_Vector_Space | [
"Topological Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Sequence",
"Definition:Cauchy Sequence/Topological Vector Space",
"Definition:Strictly Increasing/Sequence"
] | [
"Definition:Cauchy Sequence/Topological Vector Space",
"Definition:Strictly Increasing/Sequence",
"Definition:Open Neighborhood",
"Definition:Cauchy Sequence/Topological Vector Space",
"Definition:Open Neighborhood",
"Definition:Strictly Increasing/Sequence",
"Definition:Strictly Increasing/Sequence"
] |
proofwiki-22782 | Weakly Cauchy Sequence in Normed Vector Space is Bounded | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a weak Cauchy sequence in $X$.
Then $\sequence {x_n}_{n \mathop \in \N}$ is bounded. | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual of $\struct {X, \norm {\, \cdot \,}_X}$.
For each $f \in X^\ast$, $\sequence {\map f {x_n} }_{n \mathop \in \N}$ is a Cauchy sequence.
From Cauchy's Convergence Criterion, we have that $\sequence {\map f {x_n} }_{n \mathop \in \N}$ converges for e... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Weak Cauchy Sequence|weak Cauchy sequence]] in $X$.
Then $\sequence {x_n}_{n \mathop \in \N}$ is [[Definition:Bound... | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual]] of $\struct {X, \norm {\, \cdot \,}_X}$.
For each $f \in X^\ast$, $\sequence {\map f {x_n} }_{n \mathop \in \N}$ is a [[Definition:Cauchy Sequence|Cauchy sequence]].
From [[Cauchy's Convergence Criterion]], we ha... | Weakly Cauchy Sequence in Normed Vector Space is Bounded | https://proofwiki.org/wiki/Weakly_Cauchy_Sequence_in_Normed_Vector_Space_is_Bounded | https://proofwiki.org/wiki/Weakly_Cauchy_Sequence_in_Normed_Vector_Space_is_Bounded | [
"Weak Cauchy Sequences",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Weak Cauchy Sequence",
"Definition:Bounded Subset of Normed Vector Space"
] | [
"Definition:Normed Dual Space",
"Definition:Cauchy Sequence",
"Cauchy's Convergence Criterion",
"Definition:Convergent Sequence",
"Convergent Sequence in Metric Space is Bounded",
"Definition:Bounded Subset of Normed Vector Space",
"Subset of Normed Vector Space providing Bounded Point Evaluations is No... |
proofwiki-22783 | Reflexive Banach Space is Weakly Sequentially Complete | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a reflexive Banach space over $\GF$.
Then $\struct {X, \norm {\, \cdot \,}_X}$ is weakly sequentially complete. | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a weak Cauchy sequence in $X$.
From Weakly Cauchy Sequence in Normed Vector Space is Bounded, $\sequence {x_n}_{n \mathop \in \N}$ is bounded.
Then for each... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Reflexive Space|reflexive]] [[Definition:Banach Space|Banach space]] over $\GF$.
Then $\struct {X, \norm {\, \cdot \,}_X}$ is [[Definition:Weakly Sequentially Complete Topological Vector Space|weakly sequentially complete]]. | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Weak Cauchy Sequence|weak Cauchy sequence]] in $X$.
From [[Weakly Cauchy Sequence in Normed Vector Space i... | Reflexive Banach Space is Weakly Sequentially Complete | https://proofwiki.org/wiki/Reflexive_Banach_Space_is_Weakly_Sequentially_Complete | https://proofwiki.org/wiki/Reflexive_Banach_Space_is_Weakly_Sequentially_Complete | [
"Reflexive Spaces",
"Weakly Sequentially Complete Topological Vector Spaces"
] | [
"Definition:Reflexive Space",
"Definition:Banach Space",
"Definition:Weakly Sequentially Complete Topological Vector Space"
] | [
"Definition:Normed Dual Space",
"Definition:Weak Cauchy Sequence",
"Weakly Cauchy Sequence in Normed Vector Space is Bounded",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Cauchy Sequence",
"Eberlein-Šmulian Characterization of Reflexive Banach Space",
"Definition:Weak Convergence",
... |
proofwiki-22784 | Banach Space with Schur Property is Weakly Sequentially Complete | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space with the Schur property.
Then $\struct {X, \norm {\, \cdot \,}_X}$ is weakly sequentially complete. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a weak Cauchy sequence in $\struct {X, \norm {\, \cdot \,}_X}$.
From Subsequence Characterization of Cauchy Sequence in Topological Vector Space, we therefore have that:
:for all strictly increasing sequences $\sequence {n_k}_{k \mathop \in \N}$ and $\sequence {m_k}_{k \matho... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] with the [[Definition:Schur Space|Schur property]].
Then $\struct {X, \norm {\, \cdot \,}_X}$ is [[Definition:Weakly Sequentially Complete Topological Vector Space|weakly sequentially complete]]. | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Weak Cauchy Sequence|weak Cauchy sequence]] in $\struct {X, \norm {\, \cdot \,}_X}$.
From [[Subsequence Characterization of Cauchy Sequence in Topological Vector Space]], we therefore have that:
:for all [[Definition:Strictly Increasing Sequence|strictly incre... | Banach Space with Schur Property is Weakly Sequentially Complete | https://proofwiki.org/wiki/Banach_Space_with_Schur_Property_is_Weakly_Sequentially_Complete | https://proofwiki.org/wiki/Banach_Space_with_Schur_Property_is_Weakly_Sequentially_Complete | [
"Schur Spaces",
"Weakly Sequentially Complete Topological Vector Spaces"
] | [
"Definition:Banach Space",
"Definition:Schur Space",
"Definition:Weakly Sequentially Complete Topological Vector Space"
] | [
"Definition:Weak Cauchy Sequence",
"Subsequence Characterization of Cauchy Sequence in Topological Vector Space",
"Definition:Strictly Increasing/Sequence",
"Definition:Schur Space",
"Subsequence Characterization of Cauchy Sequence in Normed Vector Space",
"Definition:Cauchy Sequence",
"Definition:Banac... |
proofwiki-22785 | Rearrangement of Unconditionally Convergent Series in Banach Space Converges to Same Value | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ such that:
:$\ds \sum_{n \mathop = 1}^\infty x_n$ is unconditionally convergent.
Let:
:$\ds x = \sum_{n \mathop = 1}^\infty x_n$
Let $\pi : \N \to \N$ be a pe... | Let $\epsilon > 0$.
Take $N_1 \in \N$ such that:
:$\ds \norm {\sum_{n \mathop = 1}^N x_n - x}_X < \frac \epsilon 2$
Take $N_2 \ge N_1$ such that:
:$\ds \norm {\sum_{n \mathop \in F} x_n}_X < \frac \epsilon 4$ for every finite subset $F$ of $\set {N_2 + 1, \ldots}$.
Finally, since $\pi : \N \to \N$ is a permutation, the... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$ such that:
:$\ds \sum_{n \mathop = 1}^\infty x_n$ is [[Definition:Unconditionally Convergent Series|unco... | Let $\epsilon > 0$.
Take $N_1 \in \N$ such that:
:$\ds \norm {\sum_{n \mathop = 1}^N x_n - x}_X < \frac \epsilon 2$
Take $N_2 \ge N_1$ such that:
:$\ds \norm {\sum_{n \mathop \in F} x_n}_X < \frac \epsilon 4$ for every [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] $F$ of $\set {N_2 + 1, \ldots}$.
Fin... | Rearrangement of Unconditionally Convergent Series in Banach Space Converges to Same Value | https://proofwiki.org/wiki/Rearrangement_of_Unconditionally_Convergent_Series_in_Banach_Space_Converges_to_Same_Value | https://proofwiki.org/wiki/Rearrangement_of_Unconditionally_Convergent_Series_in_Banach_Space_Converges_to_Same_Value | [
"Unconditionally Convergent Series"
] | [
"Definition:Banach Space",
"Definition:Sequence",
"Definition:Unconditionally Convergent Series",
"Definition:Permutation"
] | [
"Definition:Finite Set",
"Definition:Subset",
"Definition:Permutation",
"Definition:Subset",
"Definition:Finite Set",
"Definition:Subset"
] |
proofwiki-22786 | Characterization of Weak Unconditional Cauchyness of Series in Banach Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $\map {c_{00} } \GF$ be the space of almost-zero sequences on $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.
{{TFAE}}
:$(1): \quad$ $\ds \sum_{n \mathop = 1}^\infty x_n$ is weakly unconditiona... | Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $\map {c_{00} } \GF$. | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\map {c_{00} } \GF$ be the [[Definition:Space of Almost-Zero Sequences|space of almost-zero sequences]] on $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|s... | Let $\norm {\, \cdot \,}_\infty$ be the [[Definition:Supremum Norm|supremum norm]] on $\map {c_{00} } \GF$. | Characterization of Weak Unconditional Cauchyness of Series in Banach Space | https://proofwiki.org/wiki/Characterization_of_Weak_Unconditional_Cauchyness_of_Series_in_Banach_Space | https://proofwiki.org/wiki/Characterization_of_Weak_Unconditional_Cauchyness_of_Series_in_Banach_Space | [
"Weakly Unconditionally Cauchy Series"
] | [
"Definition:Banach Space",
"Definition:Space of Almost-Zero Sequences",
"Definition:Sequence",
"Definition:Weakly Unconditionally Cauchy Series",
"Definition:Finite Set",
"Definition:Subset"
] | [
"Definition:Supremum Norm"
] |
proofwiki-22787 | Space of Almost-Zero Sequences is Everywhere Dense in Space of Zero-Limit Sequences | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
Let $\struct {\map {c_{00} } \GF, \norm {\, \cdot \,}_\infty}$ be the space of almost-zero sequences.
Then $\struct {\map {c_{00} } \GF, \norm {\, \cdot \,}_\infty}$ is everywhere dense in $\str... | Let $\phi \in \map {c_0} \GF$.
Let $\epsilon > 0$.
Then there exists $N \in \N$ such that:
:$\ds \sup_{n \mathop \ge N} \cmod {\map \phi n} < \epsilon$
Define $\phi_\epsilon \in \map {c_{00} } \GF$ by:
:$\ds \map {\phi_\epsilon} k = \begin{cases}\map \phi k & n \le N \\ 0 & n > N\end{cases}$
Take $n \ge N$.
We have:
:$... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Zero-Limit Sequences|space of zero-limit sequences]].
Let $\struct {\map {c_{00} } \GF, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Almost-Zero Sequences|space of almost-zero sequences]].... | Let $\phi \in \map {c_0} \GF$.
Let $\epsilon > 0$.
Then there exists $N \in \N$ such that:
:$\ds \sup_{n \mathop \ge N} \cmod {\map \phi n} < \epsilon$
Define $\phi_\epsilon \in \map {c_{00} } \GF$ by:
:$\ds \map {\phi_\epsilon} k = \begin{cases}\map \phi k & n \le N \\ 0 & n > N\end{cases}$
Take $n \ge N$.
We hav... | Space of Almost-Zero Sequences is Everywhere Dense in Space of Zero-Limit Sequences | https://proofwiki.org/wiki/Space_of_Almost-Zero_Sequences_is_Everywhere_Dense_in_Space_of_Zero-Limit_Sequences | https://proofwiki.org/wiki/Space_of_Almost-Zero_Sequences_is_Everywhere_Dense_in_Space_of_Zero-Limit_Sequences | [
"Space of Almost-Zero Sequences",
"Space of Zero-Limit Sequences"
] | [
"Definition:Space of Zero-Limit Sequences",
"Definition:Space of Almost-Zero Sequences",
"Definition:Everywhere Dense"
] | [
"Definition:Everywhere Dense",
"Category:Space of Almost-Zero Sequences",
"Category:Space of Zero-Limit Sequences"
] |
proofwiki-22788 | Space of Zero-Limit Sequences admits Schauder Basis | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
Define $e_n \in \map {c_0} \GF$ by:
:$\ds \map {e_n} k = \begin{cases}1 & n = k \\ 0 & n \ne k\end{cases}$
Then $\sequence {e_n}_{n \mathop \in \N}$ is a Schauder basis for $\struct {\map {c_0}... | For each $n \in \N$, define $e_n^\ast : \map {c_0} \GF \to \GF$ by:
:$\map {e_n^\ast} \phi = \map \phi n$
for each $\phi \in \map {c_0} \GF$.
We argue that $e_n^\ast$ is a bounded linear functional.
Let $\phi, \psi \in \map {c_0} \GF$ and $\lambda \in \GF$.
We have:
{{begin-eqn}}
{{eqn | l = \map {e_n^\ast} {\phi + \la... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Zero-Limit Sequences|space of zero-limit sequences]].
Define $e_n \in \map {c_0} \GF$ by:
:$\ds \map {e_n} k = \begin{cases}1 & n = k \\ 0 & n \ne k\end{cases}$
Then $\sequence {e_n}_{n \mathop \in ... | For each $n \in \N$, define $e_n^\ast : \map {c_0} \GF \to \GF$ by:
:$\map {e_n^\ast} \phi = \map \phi n$
for each $\phi \in \map {c_0} \GF$.
We argue that $e_n^\ast$ is a [[Definition:Bounded Linear Functional|bounded linear functional]].
Let $\phi, \psi \in \map {c_0} \GF$ and $\lambda \in \GF$.
We have:
{{begin-e... | Space of Zero-Limit Sequences admits Schauder Basis | https://proofwiki.org/wiki/Space_of_Zero-Limit_Sequences_admits_Schauder_Basis | https://proofwiki.org/wiki/Space_of_Zero-Limit_Sequences_admits_Schauder_Basis | [
"Space of Zero-Limit Sequences",
"Schauder Bases"
] | [
"Definition:Space of Zero-Limit Sequences",
"Definition:Schauder Basis"
] | [
"Definition:Bounded Linear Functional",
"Definition:Linear Functional",
"Definition:Bounded Linear Functional",
"Definition:Kronecker Delta",
"Condition for Sequence to be Schauder Basis in terms of Coordinate Functionals",
"Definition:Sequence",
"Definition:Bounded Linear Functional",
"Condition for ... |
proofwiki-22789 | Normed Dual Space of 1-Sequence Space is Isometrically Isomorphic to Space of Bounded Sequences | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$ be the $1$-sequence space.
Let $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$ be the space of bounded sequences.
Let $\struct {\map {\ell_1^\ast} \GF, \norm {\, \cdot \,}_{\ell_1^\ast} }$ be the normed dual space of $\... | Let $\sequence {e_n}_{n \mathop \in \N}$ be the Schauder basis for $\map {\ell_1} \GF$ given by P-Sequence Space admits Schauder Basis.
We first show that $T_1$ is a linear isometry.
We first argue that $f_x$ is well-defined for each $x \in \map {\ell_\infty} \GF$.
We have $\cmod {x_i} \le \norm x_\infty$ for each $i \... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$ be the [[Definition:p-Sequence Space|$1$-sequence space]].
Let $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Bounded Sequences|space of bounded sequences]].
Let $\struct {\map {\ell_1^\... | Let $\sequence {e_n}_{n \mathop \in \N}$ be the [[Definition:Schauder Basis|Schauder basis]] for $\map {\ell_1} \GF$ given by [[P-Sequence Space admits Schauder Basis]].
We first show that $T_1$ is a [[Definition:Linear Isometry|linear isometry]].
We first argue that $f_x$ is well-defined for each $x \in \map {\ell_\... | Normed Dual Space of 1-Sequence Space is Isometrically Isomorphic to Space of Bounded Sequences | https://proofwiki.org/wiki/Normed_Dual_Space_of_1-Sequence_Space_is_Isometrically_Isomorphic_to_Space_of_Bounded_Sequences | https://proofwiki.org/wiki/Normed_Dual_Space_of_1-Sequence_Space_is_Isometrically_Isomorphic_to_Space_of_Bounded_Sequences | [
"P-Sequence Spaces",
"Space of Bounded Sequences"
] | [
"Definition:p-Sequence Space",
"Definition:Space of Bounded Sequences",
"Definition:Normed Dual Space",
"Definition:Isometric Isomorphism",
"Definition:Isometric Isomorphism"
] | [
"Definition:Schauder Basis",
"P-Sequence Space admits Schauder Basis",
"Definition:Linear Isometry",
"Absolutely Convergent Series is Convergent",
"Definition:Convergent Series",
"Definition:Linear Functional",
"Linear Combination of Convergent Series",
"Definition:Linear Functional",
"Definition:Bo... |
proofwiki-22790 | Normed Dual Space of p-Sequence Space is Isometrically Isomorphic to q-Sequence Space | Let $\GF \in \set {\R, \C}$.
Let $p \in \openint 1 \infty$.
Let:
:$\ds q = \frac p {p - 1}$
so that:
:$\ds \frac 1 p + \frac 1 q = 1$
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the $p$-sequence space.
Let $\struct {\map {\ell_p^\ast} \GF, \norm {\, \cdot \,}_{\ell_p^\ast} }$ be the normed dual space of... | Let $\sequence {e_n}_{n \mathop \in \N}$ be the Schauder basis for $\map {\ell_p} \GF$ given by P-Sequence Space admits Schauder Basis.
We first show that $T_p$ is a linear isometry.
Firstly, $f_x$ is well-defined by Hölder's inequality.
Now we argue that $f_x$ is a linear functional.
Let $y, z \in \map {\ell_p} \GF$ ... | Let $\GF \in \set {\R, \C}$.
Let $p \in \openint 1 \infty$.
Let:
:$\ds q = \frac p {p - 1}$
so that:
:$\ds \frac 1 p + \frac 1 q = 1$
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the [[Definition:p-Sequence Space|$p$-sequence space]].
Let $\struct {\map {\ell_p^\ast} \GF, \norm {\, \cdot \,}_{\ell_p^... | Let $\sequence {e_n}_{n \mathop \in \N}$ be the [[Definition:Schauder Basis|Schauder basis]] for $\map {\ell_p} \GF$ given by [[P-Sequence Space admits Schauder Basis]].
We first show that $T_p$ is a [[Definition:Linear Isometry|linear isometry]].
Firstly, $f_x$ is well-defined by [[Hölder's Inequality for Sums/Formu... | Normed Dual Space of p-Sequence Space is Isometrically Isomorphic to q-Sequence Space | https://proofwiki.org/wiki/Normed_Dual_Space_of_p-Sequence_Space_is_Isometrically_Isomorphic_to_q-Sequence_Space | https://proofwiki.org/wiki/Normed_Dual_Space_of_p-Sequence_Space_is_Isometrically_Isomorphic_to_q-Sequence_Space | [
"P-Sequence Spaces",
"Normed Dual Spaces"
] | [
"Definition:p-Sequence Space",
"Definition:Normed Dual Space",
"Definition:p-Sequence Space",
"Definition:Mapping",
"Definition:Isometric Isomorphism",
"Definition:Isometric Isomorphism"
] | [
"Definition:Schauder Basis",
"P-Sequence Space admits Schauder Basis",
"Definition:Linear Isometry",
"Hölder's Inequality for Sums/Formulation 1",
"Definition:Linear Functional",
"Linear Combination of Convergent Series",
"Definition:Linear Functional",
"Definition:Bounded Linear Functional",
"Hölde... |
proofwiki-22791 | Normed Dual Space of Space of Zero-Limit Sequences is Isometrically Isomorphic to 1-Sequence Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,} }$ be the space of zero-limit sequences.
Let $\struct {\map {c_0^\ast} \GF, \norm {\, \cdot \,}_{c_0^\ast} }$ be the normed dual space of $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$.
Let $\struct {\map {\ell_1} \GF, \norm {\, \cdo... | Let $\sequence {e_n}_{n \mathop \in \N}$ be the Schauder basis for $\map {c_0} \GF$ given by Space of Zero-Limit Sequences admits Schauder Basis.
We first show that $T$ is a linear isometry.
We firstly argue that $f_x$ is well-defined for each $x \in \map {\ell_1} \GF$.
Let $y \in \map {c_0} \GF$.
We have:
:$\ds \sum_{... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,} }$ be the [[Definition:Space of Zero-Limit Sequences|space of zero-limit sequences]].
Let $\struct {\map {c_0^\ast} \GF, \norm {\, \cdot \,}_{c_0^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {\map {c_0} \G... | Let $\sequence {e_n}_{n \mathop \in \N}$ be the [[Definition:Schauder Basis|Schauder basis]] for $\map {c_0} \GF$ given by [[Space of Zero-Limit Sequences admits Schauder Basis]].
We first show that $T$ is a [[Definition:Linear Isometry|linear isometry]].
We firstly argue that $f_x$ is well-defined for each $x \in \m... | Normed Dual Space of Space of Zero-Limit Sequences is Isometrically Isomorphic to 1-Sequence Space | https://proofwiki.org/wiki/Normed_Dual_Space_of_Space_of_Zero-Limit_Sequences_is_Isometrically_Isomorphic_to_1-Sequence_Space | https://proofwiki.org/wiki/Normed_Dual_Space_of_Space_of_Zero-Limit_Sequences_is_Isometrically_Isomorphic_to_1-Sequence_Space | [
"Space of Zero-Limit Sequences",
"P-Sequence Spaces"
] | [
"Definition:Space of Zero-Limit Sequences",
"Definition:Normed Dual Space",
"Definition:P-Sequence Space",
"Definition:Mapping",
"Definition:Isometric Isomorphism",
"Definition:Isometric Isomorphism"
] | [
"Definition:Schauder Basis",
"Space of Zero-Limit Sequences admits Schauder Basis",
"Definition:Linear Isometry",
"Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach",
"Definition:Convergent Series",
"Definition:Linear Functional",
"Linear Combination of Convergent Ser... |
proofwiki-22792 | Infinite Subseries of Unconditionally Convergent Series is Unconditionally Convergent | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ such that:
:$\ds \sum_{n \mathop = 1}^\infty x_n$ is unconditionally convergent.
Let $A \subseteq \N$ be infinite.
Then:
:$\ds \sum_{n \mathop \in A} x_n = \... | From Characterization of Unconditional Convergence of Series in Banach Space:
:for each $\epsilon > 0$, there exists $n \in \N$ such that for all finite sets $F \subseteq \set {j : j \ge n + 1}$, we have $\ds \norm {\sum_{j \mathop \in F} x_j}_X < \epsilon$
From Subset of Finite Set is Finite, we can swap $F$ for $F \c... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$ such that:
:$\ds \sum_{n \mathop = 1}^\infty x_n$ is [[Definition:Unconditionally Convergent Series|unco... | From [[Characterization of Unconditional Convergence of Series in Banach Space]]:
:for each $\epsilon > 0$, there exists $n \in \N$ such that for all [[Definition:Finite Set|finite sets]] $F \subseteq \set {j : j \ge n + 1}$, we have $\ds \norm {\sum_{j \mathop \in F} x_j}_X < \epsilon$
From [[Subset of Finite Set is ... | Infinite Subseries of Unconditionally Convergent Series is Unconditionally Convergent | https://proofwiki.org/wiki/Infinite_Subseries_of_Unconditionally_Convergent_Series_is_Unconditionally_Convergent | https://proofwiki.org/wiki/Infinite_Subseries_of_Unconditionally_Convergent_Series_is_Unconditionally_Convergent | [
"Unconditionally Convergent Series"
] | [
"Definition:Banach Space",
"Definition:Sequence",
"Definition:Unconditionally Convergent Series",
"Definition:Infinite Set",
"Definition:Unconditionally Convergent Series",
"Definition:Indicator Function"
] | [
"Characterization of Unconditional Convergence of Series in Banach Space",
"Definition:Finite Set",
"Subset of Finite Set is Finite",
"Definition:Empty Set",
"Definition:Finite Set",
"Definition:Finite Set"
] |
proofwiki-22793 | Limit Superior in terms of Partitioning Subsequences | Let $\sequence {a_n}_{n \mathop \in \N}$ be a real sequence that is bounded above.
Let $\SS = \set {S_1, \ldots, S_N}$ be a finite partition of $\N$.
Let $S_j = \set {\map {\phi_j} k : k \in \N}$ where $\sequence {\map {\phi_j} k}_{k \mathop \in \N}$ is an increasing sequence.
We therefore have:
:$\ds \limsup_{k \matho... | Let $M > 0$.
We first write:
:$\ds \set {x_k : k \ge M} = \bigcup_{j \mathop = 1}^N \set {x_{\map {\phi_j} k} : \map {\phi_j} k \ge M}$
We then have, from Supremum of Union of Bounded Above Sets of Real Numbers:
:$\ds \sup_{k \ge M} x_k = \max_{1 \le j \le N} \sup_{\map {\phi_j} k \ge M} x_{\map {\phi_j} k}$
From Limit... | Let $\sequence {a_n}_{n \mathop \in \N}$ be a [[Definition:Real Sequence|real sequence]] that is [[Definition:Bounded Above Real Sequence|bounded above]].
Let $\SS = \set {S_1, \ldots, S_N}$ be a [[Definition:Finite Set|finite]] [[Definition:Set Partition|partition]] of $\N$.
Let $S_j = \set {\map {\phi_j} k : k \in ... | Let $M > 0$.
We first write:
:$\ds \set {x_k : k \ge M} = \bigcup_{j \mathop = 1}^N \set {x_{\map {\phi_j} k} : \map {\phi_j} k \ge M}$
We then have, from [[Supremum of Union of Bounded Above Sets of Real Numbers]]:
:$\ds \sup_{k \ge M} x_k = \max_{1 \le j \le N} \sup_{\map {\phi_j} k \ge M} x_{\map {\phi_j} k}$
Fro... | Limit Superior in terms of Partitioning Subsequences | https://proofwiki.org/wiki/Limit_Superior_in_terms_of_Partitioning_Subsequences | https://proofwiki.org/wiki/Limit_Superior_in_terms_of_Partitioning_Subsequences | [
"Limits Superior"
] | [
"Definition:Real Sequence",
"Definition:Bounded Above Sequence/Real",
"Definition:Finite Set",
"Definition:Set Partition",
"Definition:Increasing/Sequence",
"Definition:Limit Superior"
] | [
"Supremum of Union of Bounded Above Sets of Real Numbers",
"Limit of Maximum of Convergent Sequences is Maximum of Limits",
"Category:Limits Superior"
] |
proofwiki-22794 | Negative of Limit Superior is Limit Inferior of Negative | Let $\sequence {a_n}_{n \mathop \in \N}$ be a real sequence that is bounded above.
Then we have:
:$\ds -\limsup_{n \mathop \to \infty} a_n = \liminf_{n \mathop \to \infty} \paren {-a_n}$ | For each $N \in \N$, we have:
:$\ds -\sup_{n \ge N} a_n = \inf_{n \ge N} \paren {-a_n}$
from Negative of Supremum is Infimum of Negatives.
From Multiple Rule for Real Sequences, we have:
:$\ds -\sup_{n \ge N} a_n \to -\limsup_{n \mathop \to \infty} a_n$ as $N \to \infty$.
Also:
:$\ds \inf_{n \ge N} \paren {-a_n} \to \l... | Let $\sequence {a_n}_{n \mathop \in \N}$ be a [[Definition:Real Sequence|real sequence]] that is [[Definition:Bounded Above Real Sequence|bounded above]].
Then we have:
:$\ds -\limsup_{n \mathop \to \infty} a_n = \liminf_{n \mathop \to \infty} \paren {-a_n}$ | For each $N \in \N$, we have:
:$\ds -\sup_{n \ge N} a_n = \inf_{n \ge N} \paren {-a_n}$
from [[Negative of Supremum is Infimum of Negatives]].
From [[Multiple Rule for Real Sequences]], we have:
:$\ds -\sup_{n \ge N} a_n \to -\limsup_{n \mathop \to \infty} a_n$ as $N \to \infty$.
Also:
:$\ds \inf_{n \ge N} \paren {-a... | Negative of Limit Superior is Limit Inferior of Negative | https://proofwiki.org/wiki/Negative_of_Limit_Superior_is_Limit_Inferior_of_Negative | https://proofwiki.org/wiki/Negative_of_Limit_Superior_is_Limit_Inferior_of_Negative | [
"Limits Superior",
"Limits Inferior"
] | [
"Definition:Real Sequence",
"Definition:Bounded Above Sequence/Real"
] | [
"Negative of Supremum is Infimum of Negatives",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Convergent Real Sequence has Unique Limit",
"Category:Limits Superior",
"Category:Limits Inferior"
] |
proofwiki-22795 | Negative of Limit Inferior is Limit Superior of Negative | Let $\sequence {a_n}_{n \mathop \in \N}$ be a real sequence that is bounded below .
Then we have:
:$\ds -\liminf_{n \mathop \to \infty} a_n = \limsup_{n \mathop \to \infty} \paren {-a_n}$ | For each $N \in \N$, we have:
:$\ds -\inf_{n \ge N} a_n = \sup_{n \ge N} \paren {-a_n}$
from Negative of Infimum is Supremum of Negatives.
From Multiple Rule for Real Sequences, we have:
:$\ds -\inf_{n \ge N} a_n \to -\liminf_{n \mathop \to \infty} a_n$ as $N \to \infty$.
Also:
:$\ds \sup_{n \ge N} \paren {-a_n} \to \l... | Let $\sequence {a_n}_{n \mathop \in \N}$ be a [[Definition:Real Sequence|real sequence]] that is [[Definition:Bounded Below Real Sequence|bounded below ]].
Then we have:
:$\ds -\liminf_{n \mathop \to \infty} a_n = \limsup_{n \mathop \to \infty} \paren {-a_n}$ | For each $N \in \N$, we have:
:$\ds -\inf_{n \ge N} a_n = \sup_{n \ge N} \paren {-a_n}$
from [[Negative of Infimum is Supremum of Negatives]].
From [[Multiple Rule for Real Sequences]], we have:
:$\ds -\inf_{n \ge N} a_n \to -\liminf_{n \mathop \to \infty} a_n$ as $N \to \infty$.
Also:
:$\ds \sup_{n \ge N} \paren {-a... | Negative of Limit Inferior is Limit Superior of Negative | https://proofwiki.org/wiki/Negative_of_Limit_Inferior_is_Limit_Superior_of_Negative | https://proofwiki.org/wiki/Negative_of_Limit_Inferior_is_Limit_Superior_of_Negative | [
"Limits Superior",
"Limits Inferior"
] | [
"Definition:Real Sequence",
"Definition:Bounded Below Sequence/Real"
] | [
"Negative of Infimum is Supremum of Negatives",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Convergent Real Sequence has Unique Limit",
"Category:Limits Superior",
"Category:Limits Inferior"
] |
proofwiki-22796 | Limit Inferior in terms of Partitioning Subsequences | Let $\sequence {a_n}_{n \mathop \in \N}$ be a real sequence that is bounded below.
Let $\SS = \set {S_1, \ldots, S_N}$ be a finite partition of $\N$.
Let $S_j = \set {\map {\phi_j} k : k \in \N}$ where $\sequence {\map {\phi_j} k}_{k \mathop \in \N}$ is an increasing sequence.
We therefore have:
:$\ds \liminf_{k \matho... | We have:
{{begin-eqn}}
{{eqn | l = \liminf_{k \mathop \to \infty} x_k
| r = -\limsup_{k \mathop \to \infty} \paren {-x_k}
| c = Negative of Limit Inferior is Limit Superior of Negative
}}
{{eqn | r = -\max_{1 \le j \le N} \limsup_{k \mathop \to \infty} \paren {-x_{\map {\phi_j} k} }
}}
{{eqn | r = -\max_{1 \le j \l... | Let $\sequence {a_n}_{n \mathop \in \N}$ be a [[Definition:Real Sequence|real sequence]] that is [[Definition:Bounded Below Real Sequence|bounded below]].
Let $\SS = \set {S_1, \ldots, S_N}$ be a [[Definition:Finite Set|finite]] [[Definition:Set Partition|partition]] of $\N$.
Let $S_j = \set {\map {\phi_j} k : k \in ... | We have:
{{begin-eqn}}
{{eqn | l = \liminf_{k \mathop \to \infty} x_k
| r = -\limsup_{k \mathop \to \infty} \paren {-x_k}
| c = [[Negative of Limit Inferior is Limit Superior of Negative]]
}}
{{eqn | r = -\max_{1 \le j \le N} \limsup_{k \mathop \to \infty} \paren {-x_{\map {\phi_j} k} }
}}
{{eqn | r = -\max_{1 \le ... | Limit Inferior in terms of Partitioning Subsequences | https://proofwiki.org/wiki/Limit_Inferior_in_terms_of_Partitioning_Subsequences | https://proofwiki.org/wiki/Limit_Inferior_in_terms_of_Partitioning_Subsequences | [
"Limits Inferior",
"Limits Superior"
] | [
"Definition:Real Sequence",
"Definition:Bounded Below Sequence/Real",
"Definition:Finite Set",
"Definition:Set Partition",
"Definition:Increasing/Sequence",
"Definition:Limit Inferior"
] | [
"Negative of Limit Inferior is Limit Superior of Negative",
"Negative of Limit Inferior is Limit Superior of Negative",
"Negative of Supremum is Infimum of Negatives",
"Category:Limits Inferior",
"Category:Limits Superior"
] |
proofwiki-22797 | Real Series is Unconditionally Convergent iff Absolutely Convergent | Let $\sequence {a_n}_{n \mathop \in \N}$ be a real sequence.
Then $\ds \sum_{n \mathop = 1}^\infty a_n$ is unconditionally convergent {{iff}} it is absolutely convergent. | === Necessary Condition ===
Suppose that:
:$\ds \sum_{n \mathop = 1}^\infty a_n$ is unconditionally convergent
{{AimForCont}}:
:$\ds \sum_{n \mathop = 1}^\infty \cmod {a_n} = \infty$
From Riemann's Rearrangement Theorem, there exists a permutation $\pi : \N \to \N$ such that:
:$\ds \sum_{n \mathop = 1}^\infty a_{\map \... | Let $\sequence {a_n}_{n \mathop \in \N}$ be a [[Definition:Real Sequence|real sequence]].
Then $\ds \sum_{n \mathop = 1}^\infty a_n$ is [[Definition:Unconditionally Convergent Series|unconditionally convergent]] {{iff}} it is [[Definition:Absolutely Convergent Series|absolutely convergent]]. | === Necessary Condition ===
Suppose that:
:$\ds \sum_{n \mathop = 1}^\infty a_n$ is [[Definition:Unconditionally Convergent Series|unconditionally convergent]]
{{AimForCont}}:
:$\ds \sum_{n \mathop = 1}^\infty \cmod {a_n} = \infty$
From [[Riemann's Rearrangement Theorem]], there exists a [[Definition:Permutation|per... | Real Series is Unconditionally Convergent iff Absolutely Convergent | https://proofwiki.org/wiki/Real_Series_is_Unconditionally_Convergent_iff_Absolutely_Convergent | https://proofwiki.org/wiki/Real_Series_is_Unconditionally_Convergent_iff_Absolutely_Convergent | [
"Unconditionally Convergent Series",
"Absolutely Convergent Series"
] | [
"Definition:Real Sequence",
"Definition:Unconditionally Convergent Series",
"Definition:Absolutely Convergent Series"
] | [
"Definition:Unconditionally Convergent Series",
"Riemann's Rearrangement Theorem",
"Definition:Permutation",
"Definition:Unconditionally Convergent Series",
"Definition:Absolutely Convergent Series",
"Definition:Absolutely Convergent Series",
"Definition:Permutation",
"Definition:Unconditionally Conve... |
proofwiki-22798 | Gantmacher's Theorem | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be Banach spaces over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ and $\struct {Y^\ast, \norm {\, \cdot \,}_{Y^\ast} }$ be the respective normed dual spaces.
Let $w$ denote the weak topology.
... | Let $w$ and $w$ denote the weak topology of the relevant space. | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Banach Space|Banach spaces]] over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ and $\struct {Y^\ast, \norm {\, \cdot \,}_{Y^\ast} }$ be the respective [[Definition:Normed Dual... | Let $w$ and $w$ denote the [[Definition:Weak Topology on Topological Vector Space|weak topology]] of the relevant space. | Gantmacher's Theorem | https://proofwiki.org/wiki/Gantmacher's_Theorem | https://proofwiki.org/wiki/Gantmacher's_Theorem | [
"Weakly Compact Linear Transformations"
] | [
"Definition:Banach Space",
"Definition:Normed Dual Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Weak-* Topology",
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation",
"Definition:Evaluation Linear Transformation",
"Definition:Bounded Linear Tra... | [
"Definition:Weak Topology on Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space"
] |
proofwiki-22799 | Characterization of von Neumann-Bounded Set in Weak Topology | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,} }$.
Let $B \subseteq X$.
Then $B$ is von Neumann-bounded in $\str... | === Necessary Condition ===
Suppose that $B$ is von Neumann-bounded in $\struct {X, w}$.
Let $f \in X^\ast$.
From Open Sets in Weak Topology of Topological Vector Space:
:$V = \set {y \in X : \cmod {\map f y} < 1}$ is an open neighborhood of $\mathbf 0_X$ in $\struct {X, w}$.
Hence there exists $r > 0$ such that:
:$B \... | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w$ be the [[Definition:Weak Topology on Topological Vect... | === Necessary Condition ===
Suppose that $B$ is [[Definition:Von Neumann-Bounded Subset of Topological Vector Space|von Neumann-bounded]] in $\struct {X, w}$.
Let $f \in X^\ast$.
From [[Open Sets in Weak Topology of Topological Vector Space]]:
:$V = \set {y \in X : \cmod {\map f y} < 1}$ is an [[Definition:Open Neig... | Characterization of von Neumann-Bounded Set in Weak Topology | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Bounded_Set_in_Weak_Topology | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Bounded_Set_in_Weak_Topology | [
"Weak Topologies in Topological Vector Spaces",
"Weak Topologies on Topological Vector Spaces",
"Weak Topologies on Topological Vector Spaces",
"Von Neumann-Bounded Subsets of Topological Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Definition:Von Neumann-Bounded Subset of Topological Vector Space",
"Open Sets in Weak Topology of Topological Vector Space",
"Definition:Open Neighborhood",
"Definition:Linear Functional",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] |
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