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-22800 | Eberlein-Šmulian Characterization of Weakly Compact Linear Transformation | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be Banach spaces over $\GF$.
Let $T : X \to Y$ be a bounded linear transformation.
Then $T$ is weakly compact {{iff}}:
:for each bounded sequence $\sequence {x_n}_{n \mathop \in \N}$ in $X$, there exists a sub... | Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\cl$ be the closure taken in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\cl_w$ be the closure taken in the weak topology.
Let $B_X^-$ be the closed unit ball in $\struct {X, \norm {\, \cdot \,}_X}$. | 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 $T : X \to Y$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Then $T$ is [[Definition:Weakly Compact Linear Tr... | Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\cl$ be the [[Definition:Topological Closure|closure]] taken in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\cl_w$ be the [[Definition:Topological Closure|closure]] taken in the [[Defin... | Eberlein-Šmulian Characterization of Weakly Compact Linear Transformation | https://proofwiki.org/wiki/Eberlein-Šmulian_Characterization_of_Weakly_Compact_Linear_Transformation | https://proofwiki.org/wiki/Eberlein-Šmulian_Characterization_of_Weakly_Compact_Linear_Transformation | [
"Weakly Compact Linear Transformations"
] | [
"Definition:Banach Space",
"Definition:Bounded Linear Transformation",
"Definition:Weakly Compact Linear Transformation",
"Definition:Bounded Sequence",
"Definition:Subsequence",
"Definition:Weak Convergence"
] | [
"Definition:Weak Topology on Topological Vector Space",
"Definition:Closure (Topology)",
"Definition:Closure (Topology)",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Closed Unit Ball",
"Definition:Weak Topology on Topological Vector Space"
] |
proofwiki-22801 | Relativized 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 $\struct {X, \norm {\, \cdot \,}_X}$.
{{TFAE}}:
:$(1): \quad$ $\struct {A, w}$ is relatively compact in $\struct {X, w}$
:$(2): \quad$ $\struct {A, w}$ is sequentially compact in $\struct ... | === $(1)$ implies $(2)$ ===
Suppose that $\struct {A, w}$ is relatively compact.
Let $\sequence {a_n}_{n \mathop \in \N}$ be a sequence in $A$.
From Characterization of Separable Normed Vector Space:
:$Y = \map \cl {\map \span {\set {a_n : n \in \N} } }$ is separable.
Let $\norm {\, \cdot \,}_Y$ be the restriction of $... | 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 $\struct {X, \norm {\, \cdot \,}_X}$.
{{TFAE}}:
:$(1): \quad$ $\struct {A, w}$ is [[Definition:Rela... | === $(1)$ implies $(2)$ ===
Suppose that $\struct {A, w}$ is [[Definition:Relatively Compact|relatively compact]].
Let $\sequence {a_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $A$.
From [[Characterization of Separable Normed Vector Space]]:
:$Y = \map \cl {\map \span {\set {a_n : n \in \N} } }$ ... | Relativized Eberlein-Šmulian Theorem | https://proofwiki.org/wiki/Relativized_Eberlein-Šmulian_Theorem | https://proofwiki.org/wiki/Relativized_Eberlein-Šmulian_Theorem | [
"Weak Topologies on Topological Vector Spaces",
"Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Relatively Compact Subspace",
"Definition:Sequentially Compact Space",
"Definition:Relatively Countable Compact"
] | [
"Definition:Relatively Compact Subspace",
"Definition:Sequence",
"Characterization of Separable Normed Vector Space",
"Definition:Separable Space",
"Definition:Restriction/Mapping",
"Definition:Normed Dual Space",
"Normed Dual of Separable Normed Vector Space contains Normalized Sequence Separating Poin... |
proofwiki-22802 | Composition of Weakly Compact Linear Transformation and Bounded Linear Transformation is Weakly Compact | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$, $\struct {Y, \norm {\, \cdot \,}_Y}$ and $\struct {Z, \norm {\, \cdot \,}_Z}$ be Banach spaces over $\GF$.
Let $T : X \to Y$ be a weakly compact linear transformation.
Let $A : Y \to Z$ be a bounded linear transformation.
Let $B : Z \to X$ be a boun... | === $T B$ is weakly compact ===
By the Eberlein-Šmulian Characterization of Weakly Compact Linear Transformation, it is enough to show that:
:for each bounded sequence $\sequence {z_n}_{n \mathop \in \N}$ in $Z$, there exists a subsequence $\sequence {z_{n_k} }_{k \mathop \in \N}$ such that:
::$\sequence {T B z_{n_k} }... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$, $\struct {Y, \norm {\, \cdot \,}_Y}$ and $\struct {Z, \norm {\, \cdot \,}_Z}$ be [[Definition:Banach Space|Banach spaces]] over $\GF$.
Let $T : X \to Y$ be a [[Definition:Weakly Compact Linear Transformation|weakly compact linear transformation]].... | === $T B$ is [[Definition:Weakly Compact Linear Transformation|weakly compact]] ===
By the [[Eberlein-Šmulian Characterization of Weakly Compact Linear Transformation]], it is enough to show that:
:for each [[Definition:Bounded Sequence|bounded sequence]] $\sequence {z_n}_{n \mathop \in \N}$ in $Z$, there exists a [[D... | Composition of Weakly Compact Linear Transformation and Bounded Linear Transformation is Weakly Compact | https://proofwiki.org/wiki/Composition_of_Weakly_Compact_Linear_Transformation_and_Bounded_Linear_Transformation_is_Weakly_Compact | https://proofwiki.org/wiki/Composition_of_Weakly_Compact_Linear_Transformation_and_Bounded_Linear_Transformation_is_Weakly_Compact | [
"Weakly Compact Linear Transformations"
] | [
"Definition:Banach Space",
"Definition:Weakly Compact Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Weakly Compact Linear Transformation",
"Definition:Weakly Compact Linear Transformation"
] | [
"Definition:Weakly Compact Linear Transformation",
"Eberlein-Šmulian Characterization of Weakly Compact Linear Transformation",
"Definition:Bounded Sequence",
"Definition:Subsequence",
"Definition:Weak Convergence",
"Definition:Bounded Sequence",
"Definition:Bounded Linear Transformation",
"Definition... |
proofwiki-22803 | Unconditional Weak-* Convergence of Sum of Schauder Basis of Zero-Limit Sequences in Space of Bounded Sequences | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
Let $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$ be the $1$-sequence space.
Let $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$ be the space of bounded sequences.
Let $\se... | From Bijective Identification Mapping is Homeomorphism from Original Topology to Identification Topology:
:$T_1 : \struct {\map {\ell_\infty} \GF, w^\ast_{\ell_\infty} } \to \struct {\map {\ell_1^\ast} \GF, w^\ast}$ is a homeomorphism.
Hence it suffices to show that:
:$\ds \sum_{n \mathop = 1}^\infty f_{e_n}$ converges... | 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 $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$ be the [[Definition:P-Sequence Space|$1$-sequence space]].
Let $\struct {\map {\ell_\inft... | From [[Bijective Identification Mapping is Homeomorphism from Original Topology to Identification Topology]]:
:$T_1 : \struct {\map {\ell_\infty} \GF, w^\ast_{\ell_\infty} } \to \struct {\map {\ell_1^\ast} \GF, w^\ast}$ is a [[Definition:Homeomorphism|homeomorphism]].
Hence it suffices to show that:
:$\ds \sum_{n \mat... | Unconditional Weak-* Convergence of Sum of Schauder Basis of Zero-Limit Sequences in Space of Bounded Sequences | https://proofwiki.org/wiki/Unconditional_Weak-*_Convergence_of_Sum_of_Schauder_Basis_of_Zero-Limit_Sequences_in_Space_of_Bounded_Sequences | https://proofwiki.org/wiki/Unconditional_Weak-*_Convergence_of_Sum_of_Schauder_Basis_of_Zero-Limit_Sequences_in_Space_of_Bounded_Sequences | [
"Unconditionally Convergent Series",
"Space of Bounded Sequences",
"Weak-* Topologies"
] | [
"Definition:Space of Zero-Limit Sequences",
"Definition:P-Sequence Space",
"Definition:Space of Bounded Sequences",
"Definition:Schauder Basis",
"Space of Zero-Limit Sequences admits Schauder Basis",
"Definition:Normed Dual Space",
"Normed Dual Space of 1-Sequence Space is Isometrically Isomorphic to Sp... | [
"Bijective Identification Mapping is Homeomorphism from Original Topology to Identification Topology",
"Definition:Homeomorphism",
"Definition:Unconditionally Convergent Series",
"Definition:Permutation",
"Manipulation of Absolutely Convergent Series/Permutation",
"Definition:Absolutely Convergent Series"... |
proofwiki-22804 | Complement of Subgroup is not necessarily Unique | Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Let $K$ be a complement of $H$ in $G$.
Then it is not necessarily the case that $K$ is the only complement of $H$ in $G$. | Proof by Counterexample:
Let $\Z_2 = \set {e, a}$ be the cyclic group of order 2, where $a$ denotes the non-identity element.
Let $G = \set {e, a, b, c}$ be the Klein Four-Group
Let $H = \set {e, a}$ be a subgroup of $G$.
Let $K_1 = \set {e, b}$ be a subgroup of $G$.
Let $K_2 = \set {e, c}$ be a subgroup of $G$.
Claim:... | Let $G$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Let $K$ be a [[Definition:Complement of Subgroup|complement]] of $H$ in $G$.
Then it is not necessarily the case that $K$ is the [[Definition:Unique|only]] [[Definition:Complement of Subgroup|complement]] of $H$ in $G$. | [[Proof by Counterexample]]:
Let $\Z_2 = \set {e, a}$ be the [[Definition:Cyclic Group|cyclic group of order 2]], where $a$ denotes the non-[[Definition:Identity Element|identity element]].
Let $G = \set {e, a, b, c}$ be the [[Definition:Klein Four-Group|Klein Four-Group]]
Let $H = \set {e, a}$ be a [[Definition:Sub... | Complement of Subgroup is not necessarily Unique | https://proofwiki.org/wiki/Complement_of_Subgroup_is_not_necessarily_Unique | https://proofwiki.org/wiki/Complement_of_Subgroup_is_not_necessarily_Unique | [
"Subgroup Complements"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Complement of Subgroup",
"Definition:Unique",
"Definition:Complement of Subgroup"
] | [
"Proof by Counterexample",
"Definition:Cyclic Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Klein Four-Group",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Subgroup",
"Definition:Complement of Subgroup",
"Definition:Complement of Subgroup",
"Definiti... |
proofwiki-22805 | Intersection of Event with Complement Can't Happen | Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.
Let $A \in \Sigma$ be an events of $\EE$, so that $A \subseteq \Omega$.
Then:
:$A \cap \overline A = \O$
where $\overline A$ is the complementary event to $A$.
That is:
:$A \cap \overline A$ is an impossibility
or:
:$\map \Pr {A \cap \... | By definition:
:$A \subseteq \Omega$
and:
:$\overline A = \relcomp \Omega A$
From Intersection with Relative Complement is Empty:
:$A \cap \overline A = \O$
We then have from Probability of Empty Event is Zero that:
:$\map \Pr \Omega = 0$
The result follows by definition of impossible event.
{{qed}} | Let the [[Definition:Probability Space|probability space]] of an [[Definition:Experiment|experiment]] $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.
Let $A \in \Sigma$ be an [[Definition:Event|events]] of $\EE$, so that $A \subseteq \Omega$.
Then:
:$A \cap \overline A = \O$
where $\overline A$ is the [[Definition:Complem... | By definition:
:$A \subseteq \Omega$
and:
:$\overline A = \relcomp \Omega A$
From [[Intersection with Relative Complement is Empty]]:
:$A \cap \overline A = \O$
We then have from [[Probability of Empty Event is Zero]] that:
:$\map \Pr \Omega = 0$
The result follows by definition of [[Definition:Impossible Event|imp... | Intersection of Event with Complement Can't Happen | https://proofwiki.org/wiki/Intersection_of_Event_with_Complement_Can't_Happen | https://proofwiki.org/wiki/Intersection_of_Event_with_Complement_Can't_Happen | [
"Intersections of Events",
"Complementary Events",
"Impossible Events",
"Disjoint Events",
"Set Intersection"
] | [
"Definition:Probability Space",
"Definition:Experiment",
"Definition:Event",
"Definition:Complementary Event",
"Definition:Event/Occurrence/Impossibility",
"Definition:Disjoint Events"
] | [
"Intersection with Relative Complement is Empty",
"Probability of Empty Event is Zero",
"Definition:Event/Occurrence/Impossibility"
] |
proofwiki-22806 | Bijective Identification Mapping is Homeomorphism from Original Topology to Identification Topology | Let $\struct {X, \tau_X}$ be a topological space.
Let $f : X \to Y$ be a bijection.
Let $\tau_Y$ be the identification topology on $Y$ induced by $T$.
Then $f : \struct {X, \tau_X} \to \struct {Y, \tau_Y}$ is a homeomorphism. | From the definition of the identification topology, we have:
:$\tau_Y = \set {V \in \map \PP Y : f^{-1} \sqbrk V \in \tau_X}$
From Identification Mapping is Continuous, $f$ is continuous.
Further, let $V \in \tau_X$.
We show that $f \sqbrk V \in \tau_Y$.
From Image of Preimage under Mapping, we have that $f^{-1} \sqbrk... | Let $\struct {X, \tau_X}$ be a [[Definition:Topological Space|topological space]].
Let $f : X \to Y$ be a [[Definition:Bijection|bijection]].
Let $\tau_Y$ be the [[Definition:Identification Topology|identification topology]] on $Y$ induced by $T$.
Then $f : \struct {X, \tau_X} \to \struct {Y, \tau_Y}$ is a [[Defin... | From the definition of the [[Definition:Identification Topology|identification topology]], we have:
:$\tau_Y = \set {V \in \map \PP Y : f^{-1} \sqbrk V \in \tau_X}$
From [[Identification Mapping is Continuous]], $f$ is [[Definition:Continuous Mapping|continuous]].
Further, let $V \in \tau_X$.
We show that $f \sqbrk ... | Bijective Identification Mapping is Homeomorphism from Original Topology to Identification Topology | https://proofwiki.org/wiki/Bijective_Identification_Mapping_is_Homeomorphism_from_Original_Topology_to_Identification_Topology | https://proofwiki.org/wiki/Bijective_Identification_Mapping_is_Homeomorphism_from_Original_Topology_to_Identification_Topology | [
"Identification Topology"
] | [
"Definition:Topological Space",
"Definition:Bijection",
"Definition:Identification Topology",
"Definition:Homeomorphism"
] | [
"Definition:Identification Topology",
"Identification Mapping is Continuous",
"Definition:Continuous Mapping",
"Image of Preimage under Mapping",
"Definition:Bijection",
"Definition:Open Mapping",
"Definition:Continuous Mapping",
"Definition:Bijection",
"Definition:Homeomorphism",
"Category:Identi... |
proofwiki-22807 | Unconditionally Convergent Complex Series is Absolutely Convergent | Let $\sequence {a_n}_{n \mathop \in \N}$ be a complex sequence such that:
:$\ds \sum_{n \mathop = 1}^\infty a_n$ converges unconditionally.
Then:
:$\ds \sum_{n \mathop = 1}^\infty a_n$ converges absolutely. | Let $\pi : \N \to \N$ be a permutation.
Then:
:$\ds \sum_{n \mathop = 1}^\infty a_{\map \pi n} = \sum_{n \mathop = 1}^\infty \map \Re {a_{\map \pi n} } + i \sum_{n \mathop = 1}^\infty \map \Im {a_{\map \pi n} }$ converges.
From Convergence of Series of Complex Numbers by Real and Imaginary Part:
:$\ds \sum_{n \mathop =... | Let $\sequence {a_n}_{n \mathop \in \N}$ be a [[Definition:Complex Sequence|complex sequence]] such that:
:$\ds \sum_{n \mathop = 1}^\infty a_n$ [[Definition:Unconditionally Convergent Series|converges unconditionally]].
Then:
:$\ds \sum_{n \mathop = 1}^\infty a_n$ [[Definition:Absolutely Convergent Series|converges ... | Let $\pi : \N \to \N$ be a [[Definition:Permutation|permutation]].
Then:
:$\ds \sum_{n \mathop = 1}^\infty a_{\map \pi n} = \sum_{n \mathop = 1}^\infty \map \Re {a_{\map \pi n} } + i \sum_{n \mathop = 1}^\infty \map \Im {a_{\map \pi n} }$ [[Definition:Convergent Series|converges]].
From [[Convergence of Series of Com... | Unconditionally Convergent Complex Series is Absolutely Convergent | https://proofwiki.org/wiki/Unconditionally_Convergent_Complex_Series_is_Absolutely_Convergent | https://proofwiki.org/wiki/Unconditionally_Convergent_Complex_Series_is_Absolutely_Convergent | [
"Unconditionally Convergent Series",
"Absolutely Convergent Series"
] | [
"Definition:Complex Sequence",
"Definition:Unconditionally Convergent Series",
"Definition:Absolutely Convergent Series"
] | [
"Definition:Permutation",
"Definition:Convergent Series",
"Convergence of Series of Complex Numbers by Real and Imaginary Part",
"Definition:Convergent Series",
"Definition:Unconditionally Convergent Series",
"Real Series is Unconditionally Convergent iff Absolutely Convergent",
"Triangle Inequality/Com... |
proofwiki-22808 | Sum of Schauder Basis of Zero-Limit Sequences is Weakly Unconditionally Cauchy | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
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.
Then:
:$\ds \sum_{n \mathop = 1}^\infty e_n$ is ... | Let $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$ be the $1$-sequence space.
Let $\struct {\map {c_0^\ast} \GF, \norm {\, \cdot \,}_{c_0^\ast} }$ be the normed dual space of $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$.
We aim to show that:
:$\ds \sum_{n \mathop = 1}^\infty \cmod {\map f {x_n} } < \inf... | 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 $\sequence {e_n}_{n \mathop \in \N}$ be the [[Definition:Schauder Basis|Schauder basis]] for $\map {c_0} \GF$ given by [[Space of Zero-Limit Se... | Let $\struct {\map {\ell_1} \GF, \norm {\, \cdot \,}_1}$ be the [[Definition:P-Sequence Space|$1$-sequence space]].
Let $\struct {\map {c_0^\ast} \GF, \norm {\, \cdot \,}_{c_0^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$.
We aim to show t... | Sum of Schauder Basis of Zero-Limit Sequences is Weakly Unconditionally Cauchy | https://proofwiki.org/wiki/Sum_of_Schauder_Basis_of_Zero-Limit_Sequences_is_Weakly_Unconditionally_Cauchy | https://proofwiki.org/wiki/Sum_of_Schauder_Basis_of_Zero-Limit_Sequences_is_Weakly_Unconditionally_Cauchy | [
"Space of Zero-Limit Sequences",
"Weakly Unconditionally Cauchy Series"
] | [
"Definition:Space of Zero-Limit Sequences",
"Definition:Schauder Basis",
"Space of Zero-Limit Sequences admits Schauder Basis",
"Definition:Weakly Unconditionally Cauchy Series"
] | [
"Definition:P-Sequence Space",
"Definition:Normed Dual Space",
"Normed Dual Space of Space of Zero-Limit Sequences is Isometrically Isomorphic to 1-Sequence Space",
"Normed Dual Space of Space of Zero-Limit Sequences is Isometrically Isomorphic to 1-Sequence Space",
"Definition:Isometric Isomorphism",
"De... |
proofwiki-22809 | Series is Weakly Unconditionally Cauchy iff Linear Transformation sending Schauder Basis of Space of Zero-Limit Sequences to Terms of Sequence is Bounded | 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 $X$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
Let $\sequence {e_n}_{n \mathop \in \N}$ be the Schauder bas... | === Necessary Condition ===
Suppose that:
:$\ds \sum_{n \mathop = 1}^\infty x_n$ is weakly unconditionally Cauchy
Let $\struct {\map {c_{00} } \GF, \norm {\, \cdot \,}_\infty}$ be the space of almost-zero sequences.
Define a mapping $T : \map {c_{00} } \GF \to X$ by:
:$\ds T \xi = \sum_{n \mathop = 1}^\infty \map \xi n... | 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 $X$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Zero-Limit Seq... | === Necessary Condition ===
Suppose that:
:$\ds \sum_{n \mathop = 1}^\infty x_n$ is [[Definition:Weakly Unconditionally Cauchy Series|weakly unconditionally Cauchy]]
Let $\struct {\map {c_{00} } \GF, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Almost-Zero Sequences|space of almost-zero sequences]].
Def... | Series is Weakly Unconditionally Cauchy iff Linear Transformation sending Schauder Basis of Space of Zero-Limit Sequences to Terms of Sequence is Bounded | https://proofwiki.org/wiki/Series_is_Weakly_Unconditionally_Cauchy_iff_Linear_Transformation_sending_Schauder_Basis_of_Space_of_Zero-Limit_Sequences_to_Terms_of_Sequence_is_Bounded | https://proofwiki.org/wiki/Series_is_Weakly_Unconditionally_Cauchy_iff_Linear_Transformation_sending_Schauder_Basis_of_Space_of_Zero-Limit_Sequences_to_Terms_of_Sequence_is_Bounded | [
"Weakly Unconditionally Cauchy Series",
"Space of Zero-Limit Sequences"
] | [
"Definition:Banach Space",
"Definition:Sequence",
"Definition:Space of Zero-Limit Sequences",
"Definition:Schauder Basis",
"Space of Zero-Limit Sequences admits Schauder Basis",
"Definition:Weakly Unconditionally Cauchy Series",
"Definition:Bounded Linear Transformation"
] | [
"Definition:Weakly Unconditionally Cauchy Series",
"Definition:Space of Almost-Zero Sequences",
"Definition:Mapping",
"Definition:Linear Transformation",
"Characterization of Weak Unconditional Cauchyness of Series in Banach Space",
"Definition:Bounded Linear Transformation",
"Bounded Linear Transformat... |
proofwiki-22810 | Rearrangement of Convergent Sequence Converges to Same Limit | Let $\struct {X, \tau}$ be a topological space.
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a convergent sequence in $X$ with limit $x$.
Let $\pi : \N \to \N$ be a permutation.
Then $\sequence {x_{\map \pi n} }_{n \mathop \in \N}$ converges to $x$ in $\struct {X, \tau}$. | Let $U$ be an open neighborhood of $x$ in $\struct {X, \tau}$.
Since $\sequence {x_n}_{n \mathop \in \N}$ converges to $x$, there exists $N \in \N$ such that:
:$x_n \in U$ for $n \ge N$.
Since $\pi$ is a bijection, there exists $M \in \N$ such that:
:$\set {1, 2, \ldots, N} \subseteq \pi \sqbrk {\set {1, 2, \ldots, M} ... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Convergent Sequence|convergent sequence]] in $X$ with [[Definition:Limit of Sequence (Topology)|limit]] $x$.
Let $\pi : \N \to \N$ be a [[Definition:Permutation|... | Let $U$ be an [[Definition:Open Neighborhood|open neighborhood]] of $x$ in $\struct {X, \tau}$.
Since $\sequence {x_n}_{n \mathop \in \N}$ [[Definition:Convergent Sequence|converges to]] $x$, there exists $N \in \N$ such that:
:$x_n \in U$ for $n \ge N$.
Since $\pi$ is a [[Definition:Bijection|bijection]], there exis... | Rearrangement of Convergent Sequence Converges to Same Limit | https://proofwiki.org/wiki/Rearrangement_of_Convergent_Sequence_Converges_to_Same_Limit | https://proofwiki.org/wiki/Rearrangement_of_Convergent_Sequence_Converges_to_Same_Limit | [
"Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Convergent Sequence",
"Definition:Limit of Sequence/Topological Space",
"Definition:Permutation",
"Definition:Convergent Sequence"
] | [
"Definition:Open Neighborhood",
"Definition:Convergent Sequence",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Open Neighborhood",
"Definition:Convergent Sequence",
"Category:Topological Spaces"
] |
proofwiki-22811 | Countably Infinite Set has Uncountable Family of Subsets with Finite Intersection | Let $\CC$ be a countably infinite set.
Then there exists an uncountable family $\SS \subseteq \powerset \CC$ such that for each $A, B \in \SS$, we have that:
:$A \cap B$ is finite. | Since $\CC$ is countably infinite, there exists a bijection $f : \CC \to \N$.
From Rational Numbers are Countably Infinite, the set of rational numbers $\Q$ is countably infinite.
Hence, there exists a bijection $g : \Q \to \N$.
Then $h = f^{-1} \circ g : \Q \to \CC$ is a bijection.
We will now prove the theorem for ... | Let $\CC$ be a [[Definition:Countably Infinite Set|countably infinite set]].
Then there exists an [[Definition:Uncountable Set|uncountable]] family $\SS \subseteq \powerset \CC$ such that for each $A, B \in \SS$, we have that:
:$A \cap B$ is [[Definition:Finite Set|finite]]. | Since $\CC$ is [[Definition:Countably Infinite Set|countably infinite]], there exists a [[Definition:Bijection|bijection]] $f : \CC \to \N$.
From [[Rational Numbers are Countably Infinite]], the set of [[Definition:Rational Number|rational numbers]] $\Q$ is [[Definition:Countably Infinite Set|countably infinite]].
... | Countably Infinite Set has Uncountable Family of Subsets with Finite Intersection | https://proofwiki.org/wiki/Countably_Infinite_Set_has_Uncountable_Family_of_Subsets_with_Finite_Intersection | https://proofwiki.org/wiki/Countably_Infinite_Set_has_Uncountable_Family_of_Subsets_with_Finite_Intersection | [
"Real Analysis"
] | [
"Definition:Countably Infinite/Set",
"Definition:Uncountable/Set",
"Definition:Finite Set"
] | [
"Definition:Countably Infinite/Set",
"Definition:Bijection",
"Rational Numbers are Countably Infinite",
"Definition:Rational Number",
"Definition:Countably Infinite/Set",
"Definition:Bijection",
"Definition:Bijection",
"Rational Numbers are Everywhere Dense in Set of Real Numbers/Topology",
"Definit... |
proofwiki-22812 | Composition Mapping of Restriction Mappings | Let $S$, $T$, and $U$ be sets.
Let $f: S \to T$ and $g: T \to U$ be mappings.
Let $A \subseteq S$, $B \subseteq T$ and $C \subseteq U$.
Let $f \sqbrk S \subseteq B$ and $g \sqbrk T \subseteq C$.
Then:
{{begin-eqn}}
{{eqn| l = \paren{g \circ f} \restriction_{A \times C}
| r = \paren{g \restriction_{B \times C} } \c... | We have:
{{begin-eqn}}
{{eqn| q = \forall a \in A
| l = \map {\paren{g \circ f} \restriction_{A \times C} } a
| r = \map {g \circ f} a
| c = {{Defof|Restricted Mapping}}
}}
{{eqn| r = \map g {\map f a}
| c = {{Defof|Composite Mapping}}
}}
{{eqn| r = \map {g \restriction_{B \times C} } {\map f a}
... | Let $S$, $T$, and $U$ be [[Definition:Set|sets]].
Let $f: S \to T$ and $g: T \to U$ be [[Definition:Mapping|mappings]].
Let $A \subseteq S$, $B \subseteq T$ and $C \subseteq U$.
Let $f \sqbrk S \subseteq B$ and $g \sqbrk T \subseteq C$.
Then:
{{begin-eqn}}
{{eqn| l = \paren{g \circ f} \restriction_{A \times C}
... | We have:
{{begin-eqn}}
{{eqn| q = \forall a \in A
| l = \map {\paren{g \circ f} \restriction_{A \times C} } a
| r = \map {g \circ f} a
| c = {{Defof|Restricted Mapping}}
}}
{{eqn| r = \map g {\map f a}
| c = {{Defof|Composite Mapping}}
}}
{{eqn| r = \map {g \restriction_{B \times C} } {\map f a}
... | Composition Mapping of Restriction Mappings | https://proofwiki.org/wiki/Composition_Mapping_of_Restriction_Mappings | https://proofwiki.org/wiki/Composition_Mapping_of_Restriction_Mappings | [
"Composite Mappings",
"Restrictions"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Restriction/Mapping",
"Definition:Restriction/Mapping",
"Definition:Restriction/Mapping"
] | [
"Equality of Mappings",
"Category:Composite Mappings",
"Category:Restrictions"
] |
proofwiki-22813 | Closure of Totally Bounded Subspace is Totally Bounded | Let $\struct {M, d}$ be a metric space.
Let $A \subseteq M$ be totally bounded.
Let $\cl$ be the topological closure.
Then $\map \cl A$ is totally bounded. | Let $\epsilon > 0$.
Since $A$ is totally bounded, there exists $x_1, \ldots, x_n \in A$ such that:
:$\ds A \subseteq \bigcup_{i \mathop = 1}^n \map {B_{\epsilon/2} } {x_i}$
where $\map {B_\epsilon} {x_i}$ is the open ball of radius $\epsilon$, centered at $x_i$.
From Set Closure Preserves Set Inclusion, we have:
:$\ds ... | Let $\struct {M, d}$ be a [[Definition:Metric Space|metric space]].
Let $A \subseteq M$ be [[Definition:Totally Bounded Metric Space|totally bounded]].
Let $\cl$ be the [[Definition:Topological Closure|topological closure]].
Then $\map \cl A$ is [[Definition:Totally Bounded Metric Space|totally bounded]]. | Let $\epsilon > 0$.
Since $A$ is [[Definition:Totally Bounded Metric Space|totally bounded]], there exists $x_1, \ldots, x_n \in A$ such that:
:$\ds A \subseteq \bigcup_{i \mathop = 1}^n \map {B_{\epsilon/2} } {x_i}$
where $\map {B_\epsilon} {x_i}$ is the [[Definition:Open Ball|open ball]] of [[Definition:Radius of Op... | Closure of Totally Bounded Subspace is Totally Bounded | https://proofwiki.org/wiki/Closure_of_Totally_Bounded_Subspace_is_Totally_Bounded | https://proofwiki.org/wiki/Closure_of_Totally_Bounded_Subspace_is_Totally_Bounded | [
"Totally Bounded Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Totally Bounded Metric Space",
"Definition:Closure (Topology)",
"Definition:Totally Bounded Metric Space"
] | [
"Definition:Totally Bounded Metric Space",
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Set Closure Preserves Set Inclusion",
"Closure of Finite Union equals Union of Closures",
"Closure of Open Ball in Metric Space",
"Definition:Closed Ball",
"Definition:To... |
proofwiki-22814 | Subspace of Complete Metric Space is Relatively Compact iff Totally Bounded | Let $\struct {M, d}$ be a metric space.
Let $A \subseteq M$.
We have that $A$ is relatively compact in $\struct {M, d}$ {{iff}} $A$ is totally bounded. | Let $\cl$ be the topological closure. | Let $\struct {M, d}$ be a [[Definition:Complete Metric Space|metric space]].
Let $A \subseteq M$.
We have that $A$ is [[Definition:Relatively Compact Subspace|relatively compact]] in $\struct {M, d}$ {{iff}} $A$ is [[Definition:Totally Bounded Metric Space|totally bounded]]. | Let $\cl$ be the [[Definition:Topological Closure|topological closure]]. | Subspace of Complete Metric Space is Relatively Compact iff Totally Bounded | https://proofwiki.org/wiki/Subspace_of_Complete_Metric_Space_is_Relatively_Compact_iff_Totally_Bounded | https://proofwiki.org/wiki/Subspace_of_Complete_Metric_Space_is_Relatively_Compact_iff_Totally_Bounded | [
"Complete Metric Spaces",
"Totally Bounded Metric Spaces"
] | [
"Definition:Complete Metric Space",
"Definition:Relatively Compact Subspace",
"Definition:Totally Bounded Metric Space"
] | [
"Definition:Closure (Topology)"
] |
proofwiki-22815 | Unconditionally Convergent Series in Banach Space is Weakly Unconditionally Cauchy | 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.
Then:
:$\ds \sum_{n \mathop = 1}^\infty x_n$ is weakly unconditionally Cauchy... | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $f \in X^\ast$.
Let $\pi : \N \to \N$ be a permutation.
Then:
:$\ds \sum_{n \mathop = 1}^\infty x_{\map \pi n}$ converges.
We then have:
{{begin-eqn}}
{{eqn | l = \map f {\sum_{n \mathop = 1}^\in... | 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 $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $f \in X^\ast$.
Let $\pi : \N \to \N$ be a [[Definition:Permutation|permutation]].
Then:
:$\ds \sum_{n \mathop = 1}^\infty x_{\map \pi n}$ [[Definition:Converg... | Unconditionally Convergent Series in Banach Space is Weakly Unconditionally Cauchy | https://proofwiki.org/wiki/Unconditionally_Convergent_Series_in_Banach_Space_is_Weakly_Unconditionally_Cauchy | https://proofwiki.org/wiki/Unconditionally_Convergent_Series_in_Banach_Space_is_Weakly_Unconditionally_Cauchy | [
"Unconditionally Convergent Series",
"Weakly Unconditionally Cauchy Series"
] | [
"Definition:Banach Space",
"Definition:Sequence",
"Definition:Unconditionally Convergent Series",
"Definition:Weakly Unconditionally Cauchy Series"
] | [
"Definition:Normed Dual Space",
"Definition:Permutation",
"Definition:Convergent Series",
"Continuous Mappings preserve Convergent Sequences",
"Definition:Unconditionally Convergent Series",
"Unconditionally Convergent Complex Series is Absolutely Convergent",
"Definition:Weakly Unconditionally Cauchy S... |
proofwiki-22816 | Weakly Open Set in Hausdorff Topological Vector Space is Open | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$ with weak topology $w$.
Let $U$ be an open set in $\struct {X, w}$.
Then $U$ is open in $\struct {X, \tau}$. | Let $X^\ast$ be the topological dual of $X$.
From the definition of the weak topology, $w$ is the initial topology on $X$ generated by $X^\ast$.
Since every function in $X^\ast$ is continuous, we have that $w \subseteq \tau$ from Domain Topology Contains Initial Topology iff Mappings are Continuous.
{{qed}}
Category:T... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Hausdorff Topological Vector Space|topological vector space]] over $\GF$ with [[Definition:Weak Topology on Topological Vector Space|weak topology]] $w$.
Let $U$ be an [[Definition:Open Set|open set]] in $\struct {X, w}$.
Then $U$ is [[Definiti... | Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual]] of $X$.
From the definition of the [[Definition:Weak Topology on Topological Vector Space|weak topology]], $w$ is the [[Definition:Initial Topology|initial topology]] on $X$ generated by $X^\ast$.
Since every [[Definition:Function|function]] ... | Weakly Open Set in Hausdorff Topological Vector Space is Open | https://proofwiki.org/wiki/Weakly_Open_Set_in_Hausdorff_Topological_Vector_Space_is_Open | https://proofwiki.org/wiki/Weakly_Open_Set_in_Hausdorff_Topological_Vector_Space_is_Open | [
"Topological Vector Spaces",
"Weakly Open Sets"
] | [
"Definition:Hausdorff Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Open Set",
"Definition:Open Set"
] | [
"Definition:Topological Dual Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Initial Topology",
"Definition:Function",
"Definition:Continuous Mapping",
"Domain Topology Contains Initial Topology iff Mappings are Continuous",
"Category:Topological Vector Spaces",
"Category:W... |
proofwiki-22817 | Weak-* Open Set in Normed Dual Space is Open | 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 space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w^\ast$ be a weak-$\ast$ topology on $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$.
... | 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.
From the definition of the weak-$\ast$ topology, $w^\ast$ is the initial topology on $X^{\ast \ast}$ generate... | 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 space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w^\ast$ be a [[De... | 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]].
From the definition of th... | Weak-* Open Set in Normed Dual Space is Open | https://proofwiki.org/wiki/Weak-*_Open_Set_in_Normed_Dual_Space_is_Open | https://proofwiki.org/wiki/Weak-*_Open_Set_in_Normed_Dual_Space_is_Open | [
"Normed Dual Spaces",
"Weak-* Topologies",
"Normed Dual Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Definition:Open Set",
"Definition:Open Set"
] | [
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation",
"Definition:Weak-* Topology",
"Definition:Initial Topology",
"Definition:Second Normed Dual",
"Definition:Continuous Mapping",
"Domain Topology Contains Initial Topology iff Mappings are Continuous",
"Definition:Topology",... |
proofwiki-22818 | Weak-* Topology and Norm Topology Coincide on Norm Compact Subsets of Normed Dual Space | Let $\GF \in \set {\R, \C}$.
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 $w^\ast$ be the weak-$\ast$ topology on $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$.
Let $\str... | We show that the identity mapping $\iota : \struct {K, \norm {\, \cdot \,}_{X^\ast} } \to \struct {K, w^\ast}$ is a homeomorphism.
From Weak-* Open Set in Normed Dual Space is Open, every open set in $\struct {K, w^\ast}$ is open in $\struct {K, \norm {\, \cdot \,}_{X^\ast} }$.
Hence $\iota : \struct {K, \norm {\, \cdo... | Let $\GF \in \set {\R, \C}$.
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 $w^\ast$ be the [[Definition:... | We show that the [[Definition:Identity Mapping|identity mapping]] $\iota : \struct {K, \norm {\, \cdot \,}_{X^\ast} } \to \struct {K, w^\ast}$ is a [[Definition:Homeomorphism|homeomorphism]].
From [[Weak-* Open Set in Normed Dual Space is Open]], every [[Definition:Open Set|open set]] in $\struct {K, w^\ast}$ is [[Def... | Weak-* Topology and Norm Topology Coincide on Norm Compact Subsets of Normed Dual Space | https://proofwiki.org/wiki/Weak-*_Topology_and_Norm_Topology_Coincide_on_Norm_Compact_Subsets_of_Normed_Dual_Space | https://proofwiki.org/wiki/Weak-*_Topology_and_Norm_Topology_Coincide_on_Norm_Compact_Subsets_of_Normed_Dual_Space | [
"Weak-* Topologies"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Definition:Compact Topological Space",
"Definition:Open Set"
] | [
"Definition:Identity Mapping",
"Definition:Homeomorphism",
"Weak-* Open Set in Normed Dual Space is Open",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Continuous Mapping",
"Definition:Bijection",
"Weak-* Topology is Hausdorff",
"Definition:T2 Space",
"Definition:Compact Topological S... |
proofwiki-22819 | Weak Topology and Original Topology Coincide on Originally Compact Subsets of Hausdorff Topological Vector Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a Hausdorff topological vector space over $\GF$ with weak topology $w$.
Let $\struct {K, \tau}$ be compact.
Let $U \subseteq X$.
Then $U$ is open in $\struct {K, \tau}$ {{iff}} $U$ is open in $\struct {K, w}$. | We show that the identity mapping $\iota : \struct {K, \tau} \to \struct {K, w}$ is a homeomorphism.
From Weakly Open Set in Hausdorff Topological Vector Space is Open, every open set in $\struct {K, w}$ is open in $\struct {K, \tau}$.
Hence $\iota : \struct {K, \norm {\, \cdot \,}_{X^\ast} } \to \struct {K, w^\ast}$ i... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Hausdorff Topological Vector Space|Hausdorff topological vector space]] over $\GF$ with [[Definition:Weak Topology on Topological Vector Space|weak topology]] $w$.
Let $\struct {K, \tau}$ be [[Definition:Compact Topological Space|compact]].
Let ... | We show that the [[Definition:Identity Mapping|identity mapping]] $\iota : \struct {K, \tau} \to \struct {K, w}$ is a [[Definition:Homeomorphism|homeomorphism]].
From [[Weakly Open Set in Hausdorff Topological Vector Space is Open]], every [[Definition:Open Set|open set]] in $\struct {K, w}$ is [[Definition:Open Set|o... | Weak Topology and Original Topology Coincide on Originally Compact Subsets of Hausdorff Topological Vector Space | https://proofwiki.org/wiki/Weak_Topology_and_Original_Topology_Coincide_on_Originally_Compact_Subsets_of_Hausdorff_Topological_Vector_Space | https://proofwiki.org/wiki/Weak_Topology_and_Original_Topology_Coincide_on_Originally_Compact_Subsets_of_Hausdorff_Topological_Vector_Space | [
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Hausdorff Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Compact Topological Space",
"Definition:Open Set",
"Definition:Open Set"
] | [
"Definition:Identity Mapping",
"Definition:Homeomorphism",
"Weakly Open Set in Hausdorff Topological Vector Space is Open",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Continuous Mapping",
"Definition:Bijection",
"Weak Topology on Topological Vector Space over Hausdorff Topological Field... |
proofwiki-22820 | Compact Linear Transformation is Weakly Compact | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be Banach spaces over $\GF$.
Let $T : X \to Y$ be compact linear transformation.
Then $T$ is weakly compact. | Let $B_X^-$ be the closed unit ball in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\cl$ be the topological closure taken in $\struct {X, \norm {\, \cdot \,}_X}$.
Since $T$ is compact, $\map \cl {T \sqbrk {B_X^-} }$ is compact.
From Weak Topology and Original Topology Coincide on Originally Compact Subsets of Hausdorff T... | 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 $T : X \to Y$ be [[Definition:Compact Linear Transformation|compact linear transformation]].
Then $T$ is [[Definition:Weakly Compact Linear Tran... | Let $B_X^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\cl$ be the [[Definition:Topological Closure|topological closure]] taken in $\struct {X, \norm {\, \cdot \,}_X}$.
Since $T$ is [[Definition:Compact Linear Transformation|compact]], $\map \cl {T \sqbrk {B_... | Compact Linear Transformation is Weakly Compact | https://proofwiki.org/wiki/Compact_Linear_Transformation_is_Weakly_Compact | https://proofwiki.org/wiki/Compact_Linear_Transformation_is_Weakly_Compact | [
"Weakly Compact Linear Transformations",
"Compact Linear Transformations"
] | [
"Definition:Banach Space",
"Definition:Compact Linear Transformation",
"Definition:Weakly Compact Linear Transformation"
] | [
"Definition:Closed Unit Ball",
"Definition:Closure (Topology)",
"Definition:Compact Linear Transformation",
"Definition:Compact Topological Space",
"Weak Topology and Original Topology Coincide on Originally Compact Subsets of Hausdorff Topological Vector Space",
"Definition:Weakly Compact Set",
"Defini... |
proofwiki-22821 | Subset of Totally Bounded Metric Space is Totally Bounded | Let $\struct {M, d}$ be a totally bounded metric space.
Let $A \subseteq M$.
Then $A$ is totally bounded. | Let $\epsilon > 0$.
Since $M$ is totally bounded, there exists $x_1, \ldots, x_n \in M$ such that:
:$\ds M = \bigcup_{j \mathop = 1}^n \map {B_{\epsilon/2} } {x_j}$
where $\map {B_{\epsilon/2} } {x_j}$ is the open ball with radius $\epsilon/2$ and center $x_i$.
We have:
:$\ds A \subseteq \bigcup_{j \mathop = 1}^n \map... | Let $\struct {M, d}$ be a [[Definition:Totally Bounded Metric Space|totally bounded metric space]].
Let $A \subseteq M$.
Then $A$ is [[Definition:Totally Bounded Metric Space|totally bounded]]. | Let $\epsilon > 0$.
Since $M$ is [[Definition:Totally Bounded Metric Space|totally bounded]], there exists $x_1, \ldots, x_n \in M$ such that:
:$\ds M = \bigcup_{j \mathop = 1}^n \map {B_{\epsilon/2} } {x_j}$
where $\map {B_{\epsilon/2} } {x_j}$ is the [[Definition:Open Ball|open ball]] with [[Definition:Radius of Ope... | Subset of Totally Bounded Metric Space is Totally Bounded | https://proofwiki.org/wiki/Subset_of_Totally_Bounded_Metric_Space_is_Totally_Bounded | https://proofwiki.org/wiki/Subset_of_Totally_Bounded_Metric_Space_is_Totally_Bounded | [
"Totally Bounded Metric Spaces"
] | [
"Definition:Totally Bounded Metric Space",
"Definition:Totally Bounded Metric Space"
] | [
"Definition:Totally Bounded Metric Space",
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Definition:Set Union",
"Definition:Totally Bounded Metric Space",
"Category:Totally Bounded Metric Spaces"
] |
proofwiki-22822 | Complex Plane is Separable | Let $\struct {\C, \tau_d}$ be the complex number line with the usual (Euclidean) topology.
Then $\struct {\C, \tau_d}$ is separable. | {{Recall|Separable Space}}
{{:Definition:Separable Space}}
Consider the set:
:$\Q + i \Q = \set {\alpha + i \beta : \alpha, \beta \in \Q}$
which is trivially a subset of $\C$.
We first show that $\Q + i \Q$ is countable.
Define $f : \Q^2 \to \Q + i \Q$ by:
:$\map f {x, y} = x + i y$ for each $x, y \in \Q$.
This funct... | Let $\struct {\C, \tau_d}$ be the [[Definition:Euclidean Space/Euclidean Topology/Complex|complex number line with the usual (Euclidean) topology]].
Then $\struct {\C, \tau_d}$ is [[Definition:Separable Space|separable]]. | {{Recall|Separable Space}}
{{:Definition:Separable Space}}
Consider the [[Definition:Set|set]]:
:$\Q + i \Q = \set {\alpha + i \beta : \alpha, \beta \in \Q}$
which is trivially a [[Definition:Subset|subset]] of $\C$.
We first show that $\Q + i \Q$ is [[Definition:Countable Set|countable]].
Define $f : \Q^2 \to \Q ... | Complex Plane is Separable | https://proofwiki.org/wiki/Complex_Plane_is_Separable | https://proofwiki.org/wiki/Complex_Plane_is_Separable | [
"Complex Plane",
"Separable Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Complex",
"Definition:Separable Space"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Countable Set",
"Definition:Surjection",
"Definition:Injection",
"Rational Numbers are Countably Infinite",
"Definition:Countably Infinite/Set",
"Cartesian Product of Countable Sets is Countable",
"Definition:Countably Infinite/Set",
"Definition:B... |
proofwiki-22823 | Quotient of Banach Space by Topologically Complemented Subspace is Linearly Isomorphic to Topological Complement | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $Y$ be a closed vector subspace of $X$ that is topologically complemented.
From Closed Subspace of Banach Space is Topologically Complemented iff has Topological Complement, $Y$ has a topological complement $Z$.
Let ... | Let $\pi : X \to X/Y$ be the quotient mapping.
Let $\pi_Z : Z \to X/Y$ be the restriction of $\pi$ to $Z$.
From Quotient Mapping is Bounded in Normed Quotient Vector Space, $\pi_Z$ is a bounded linear transformation.
We argue that $\pi_Z$ is our desired linear isomorphism.
For this, we will apply the Banach Isomorphism... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $Y$ be a [[Definition:Closed Set|closed]] [[Definition:Vector Subspace|vector subspace]] of $X$ that is [[Definition:Topologically Complemented Subspace of Banach Space|topologically com... | Let $\pi : X \to X/Y$ be the [[Definition:Quotient Mapping|quotient mapping]].
Let $\pi_Z : Z \to X/Y$ be the [[Definition:Restriction of Mapping|restriction]] of $\pi$ to $Z$.
From [[Quotient Mapping is Bounded in Normed Quotient Vector Space]], $\pi_Z$ is a [[Definition:Bounded Linear Transformation|bounded linear ... | Quotient of Banach Space by Topologically Complemented Subspace is Linearly Isomorphic to Topological Complement | https://proofwiki.org/wiki/Quotient_of_Banach_Space_by_Topologically_Complemented_Subspace_is_Linearly_Isomorphic_to_Topological_Complement | https://proofwiki.org/wiki/Quotient_of_Banach_Space_by_Topologically_Complemented_Subspace_is_Linearly_Isomorphic_to_Topological_Complement | [
"Normed Quotient Vector Spaces",
"Topologically Complemented Subspaces of Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Closed Set",
"Definition:Vector Subspace",
"Definition:Topologically Complemented Subspace of Banach Space",
"Closed Subspace of Banach Space is Topologically Complemented iff has Topological Complement",
"Definition:Topological Complement of Closed Vector Subspace o... | [
"Definition:Quotient Mapping",
"Definition:Restriction/Mapping",
"Quotient Mapping is Bounded in Normed Quotient Vector Space",
"Definition:Bounded Linear Transformation",
"Definition:Linear Isomorphism",
"Banach Isomorphism Theorem",
"Definition:Injection",
"Definition:Linear Transformation",
"Kern... |
proofwiki-22824 | Linear Isomorphic Image of Complemented Subspace of Banach Space is Complemented | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be Banach spaces over $\GF$ such that:
:there exists a linear isomorphism $T : X \to Y$.
Let $X_1$ be a complemented subspace of $X$ with topological complement $X_2$.
Then $T \sqbrk {X_1}$ is a complemented ... | By the definition of a complemented subspace, we have:
:$X = X_1 + X_2$
with:
:$X_1 \cap X_2 = \set { {\mathbf 0}_X}$
We show that:
:$Y = T \sqbrk {X_1} + T \sqbrk {X_2}$
and:
:$T \sqbrk {X_1} \cap T \sqbrk {X_2} = \set { {\mathbf 0}_Y}$
Let $y \in Y$.
Since $T$ is a linear isomorphism, there exists $x \in X$ such that... | 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 $\GF$ such that:
:there exists a [[Definition:Linear Isomorphism|linear isomorphism]] $T : X \to Y$.
Let $X_1$ be a [[Definition:Topologically Complemented ... | By the definition of a [[Definition:Topologically Complemented Subspace of Banach Space|complemented subspace]], we have:
:$X = X_1 + X_2$
with:
:$X_1 \cap X_2 = \set { {\mathbf 0}_X}$
We show that:
:$Y = T \sqbrk {X_1} + T \sqbrk {X_2}$
and:
:$T \sqbrk {X_1} \cap T \sqbrk {X_2} = \set { {\mathbf 0}_Y}$
Let $y \in Y$... | Linear Isomorphic Image of Complemented Subspace of Banach Space is Complemented | https://proofwiki.org/wiki/Linear_Isomorphic_Image_of_Complemented_Subspace_of_Banach_Space_is_Complemented | https://proofwiki.org/wiki/Linear_Isomorphic_Image_of_Complemented_Subspace_of_Banach_Space_is_Complemented | [
"Topologically Complete Spaces",
"Topologically Complemented Subspaces of Banach Spaces",
"Topologically Complemented Subspaces of Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Linear Isomorphism",
"Definition:Topologically Complemented Subspace of Banach Space",
"Definition:Topological Complement of Closed Vector Subspace of Banach Space",
"Definition:Topologically Complemented Subspace of Banach Space",
"Definition:Topological Complement ... | [
"Definition:Topologically Complemented Subspace of Banach Space",
"Definition:Linear Isomorphism",
"Definition:Linear Transformation",
"Image of Intersection under Injection",
"Definition:Internal Direct Sum of Modules"
] |
proofwiki-22825 | Normalized Block Basic Sequence of Schauder Basis in P-Sequence Space is Isometrically Equivalent to Schauder Basis | Let $\GF \in \set {\R, \C}$.
Let $p \in \hointr 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the $p$-sequence space.
Let $\sequence {e_n}_{n \mathop \in \N}$ be the Schauder basis given by P-Sequence Space admits Schauder Basis.
Let $\sequence {y_k}_{k \mathop \in \N}$ be a block basic sequenc... | Let $\sequence {r_j}_{j \mathop \in \N}$ be a strictly increasing sequence in $\N$ and $\sequence {a_j}_{j \mathop \in \N}$ be a sequence in $\GF$ such that:
:$\ds y_k = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} a_j e_j$
where $r_0 = 0$.
By assumption, we have:
:$\ds 1 = \norm {y_k}_p^p = \sum_{j \mathop = r_{k - 1} + 1}^... | Let $\GF \in \set {\R, \C}$.
Let $p \in \hointr 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the [[Definition:P-Sequence Space|$p$-sequence space]].
Let $\sequence {e_n}_{n \mathop \in \N}$ be the [[Definition:Schauder Basis|Schauder basis]] given by [[P-Sequence Space admits Schauder Basis... | Let $\sequence {r_j}_{j \mathop \in \N}$ be a [[Definition:Strictly Increasing Sequence|strictly increasing sequence]] in $\N$ and $\sequence {a_j}_{j \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\GF$ such that:
:$\ds y_k = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} a_j e_j$
where $r_0 = 0$.
By assumption, w... | Normalized Block Basic Sequence of Schauder Basis in P-Sequence Space is Isometrically Equivalent to Schauder Basis | https://proofwiki.org/wiki/Normalized_Block_Basic_Sequence_of_Schauder_Basis_in_P-Sequence_Space_is_Isometrically_Equivalent_to_Schauder_Basis | https://proofwiki.org/wiki/Normalized_Block_Basic_Sequence_of_Schauder_Basis_in_P-Sequence_Space_is_Isometrically_Equivalent_to_Schauder_Basis | [
"Basic Sequences",
"Block Basic Sequences",
"Block Basic Sequences",
"P-Sequence Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:Schauder Basis",
"P-Sequence Space admits Schauder Basis",
"Definition:Block Basic Sequence",
"Definition:Isometric Equivalence of Basic Sequences"
] | [
"Definition:Strictly Increasing/Sequence",
"Definition:Sequence",
"Definition:Space of Almost-Zero Sequences",
"Characterization of Isometric Equivalence of Basic Sequences",
"Characterization of Isometric Equivalence of Basic Sequences",
"Definition:Isometric Equivalence of Basic Sequences"
] |
proofwiki-22826 | Inverse Image Mapping of Identity is Identity | Let $S$ be a set.
Then:
:$\paren{I_S}^\gets = I_{\powerset S}$
where:
:$I_S$ denotes the identity mapping on $S$
:$\paren{I_S}^\gets$ denotes the inverse image mapping of $I_S$
:$\powerset S$ denotes the powerset of $S$
:$I_{\powerset S}$ denotes the identity mapping on $\powerset S$ | We have:
{{begin-eqn}}
{{eqn | l = \paren{I_S}^\gets
| r = \paren{\paren{I_S}^\to}^{-1}
| c = Identity Mapping is Bijection, Inverse Image Mapping of Bijection is Inverse of Direct Image Mapping
}}
{{eqn | r = \paren{I_{\powerset S} }^{-1}
| c = Direct Image Mapping of Identity is Identity
}}
{{eqn | ... | Let $S$ be a [[Definition:Set|set]].
Then:
:$\paren{I_S}^\gets = I_{\powerset S}$
where:
:$I_S$ denotes the [[Definition:Identity Mapping|identity mapping]] on $S$
:$\paren{I_S}^\gets$ denotes the [[Definition:Inverse Image Mapping|inverse image mapping]] of $I_S$
:$\powerset S$ denotes the [[Definition:Powerset|powe... | We have:
{{begin-eqn}}
{{eqn | l = \paren{I_S}^\gets
| r = \paren{\paren{I_S}^\to}^{-1}
| c = [[Identity Mapping is Bijection]], [[Inverse Image Mapping of Bijection is Inverse of Direct Image Mapping]]
}}
{{eqn | r = \paren{I_{\powerset S} }^{-1}
| c = [[Direct Image Mapping of Identity is Identity]]... | Inverse Image Mapping of Identity is Identity | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Identity_is_Identity | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Identity_is_Identity | [
"Inverse Image Mappings",
"Identity Mappings",
"Power Set"
] | [
"Definition:Set",
"Definition:Identity Mapping",
"Definition:Inverse Image Mapping",
"Definition:Power Set",
"Definition:Identity Mapping"
] | [
"Identity Mapping is Bijection",
"Inverse Image Mapping of Bijection is Inverse of Direct Image Mapping",
"Direct Image Mapping of Identity is Identity",
"Inverse of Identity Mapping",
"Category:Inverse Image Mappings",
"Category:Identity Mappings",
"Category:Power Set"
] |
proofwiki-22827 | Restriction of Identity Mapping is Identity Mapping | Let $S$ be a set.
Let $A$ be a subset of $S$.
Then:
:$I_S \restriction_{A \times A} = I_A$
where:
:$I_S$ denotes the identity mapping on $S$
:$I_S \restriction_{A \times A}$ denotes the restriction of $I_S$ to $A \times A$
:$I_A$ denotes the identity mapping on $A$ | We have:
{{begin-eqn}}
{{eqn | q = \forall a \in A
| l = \map {\paren{I_S \restriction_{A \times A} } } a
| r = \map {I_S} a
| c = {{Defof|Restricted Mapping}}
}}
{{eqn | r = a
| c = {{Defof|Identity Mapping}}
}}
{{eqn | r = \map {I_A} a
| c = {{Defof|Identity Mapping}}
}}
{{end-eqn}}
From... | Let $S$ be a [[Definition:Set|set]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Then:
:$I_S \restriction_{A \times A} = I_A$
where:
:$I_S$ denotes the [[Definition:Identity Mapping|identity mapping]] on $S$
:$I_S \restriction_{A \times A}$ denotes the [[Definition:Restricted Mapping|restriction]] of $I_S$ to $... | We have:
{{begin-eqn}}
{{eqn | q = \forall a \in A
| l = \map {\paren{I_S \restriction_{A \times A} } } a
| r = \map {I_S} a
| c = {{Defof|Restricted Mapping}}
}}
{{eqn | r = a
| c = {{Defof|Identity Mapping}}
}}
{{eqn | r = \map {I_A} a
| c = {{Defof|Identity Mapping}}
}}
{{end-eqn}}
Fr... | Restriction of Identity Mapping is Identity Mapping | https://proofwiki.org/wiki/Restriction_of_Identity_Mapping_is_Identity_Mapping | https://proofwiki.org/wiki/Restriction_of_Identity_Mapping_is_Identity_Mapping | [
"Identity Mappings",
"Restrictions"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Identity Mapping",
"Definition:Restriction/Mapping",
"Definition:Identity Mapping"
] | [
"Equality of Mappings",
"Category:Identity Mappings",
"Category:Restrictions"
] |
proofwiki-22828 | Characterization of Isometric Equivalence of Basic Sequences | 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 $X$.
Let $\sequence {y_n}_{n \mathop \in \N}$ be a basic sequence in $Y$.
Let $\struct {\map \BB {\sqbrk {x_n}_{n \... | We have that $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ are isometrically equivalent {{iff}}:
:there exists an isometric isomorphism $T : \sqbrk {x_n}_{n \mathop \in \N} \to \sqbrk {y_n}_{n \mathop \in \N}$ such that:
::$T x_n = y_n$ for each $n \in \N$.
From Invertible Bounded Linear... | 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 $X$.
Let $\sequence {y_n}_{n \mathop \in \N}$ be a [[De... | We have that $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ are [[Definition:Isometric Equivalence of Basic Sequences|isometrically equivalent]] {{iff}}:
:there exists an [[Definition:Isometric Isomorphism|isometric isomorphism]] $T : \sqbrk {x_n}_{n \mathop \in \N} \to \sqbrk {y_n}_{n \m... | Characterization of Isometric Equivalence of Basic Sequences | https://proofwiki.org/wiki/Characterization_of_Isometric_Equivalence_of_Basic_Sequences | https://proofwiki.org/wiki/Characterization_of_Isometric_Equivalence_of_Basic_Sequences | [
"Basic Sequences"
] | [
"Definition:Banach Space",
"Definition:Basic Sequence",
"Definition:Basic Sequence",
"Definition:Space of Bounded Linear Transformations",
"Definition:Space of Almost-Zero Sequences",
"Definition:Isometric Equivalence of Basic Sequences"
] | [
"Definition:Isometric Equivalence of Basic Sequences",
"Definition:Isometric Isomorphism",
"Invertible Bounded Linear Transformation with Norm and Norm of Inverse Bounded Above by One is Isometric Isomorphism",
"Definition:Isometric Isomorphism",
"Basic Sequences in Banach Spaces are Equivalent iff Map betw... |
proofwiki-22829 | There Exist a Countable Number of Computable Numbers | The computable numbers form a countably infinite set. | Let $S$ be the set of computable numbers.
From Rational Number is Computable, we know that $S$ is at least countably infinite.
{{Recall|Computable Number|computable number}}
{{:Definition:Computable Number}}
Such an algorithm can be implemented by a computer program.
The set of computer programs that compute computable... | The [[Definition:Computable Number|computable numbers]] form a [[Definition:Countably Infinite Set|countably infinite set]]. | Let $S$ be the [[Definition:Set|set]] of [[Definition:Computable Number|computable numbers]].
From [[Rational Number is Computable]], we know that $S$ is at least [[Definition:Countably Infinite Set|countably infinite]].
{{Recall|Computable Number|computable number}}
{{:Definition:Computable Number}}
Such an [[Defi... | There Exist a Countable Number of Computable Numbers | https://proofwiki.org/wiki/There_Exist_a_Countable_Number_of_Computable_Numbers | https://proofwiki.org/wiki/There_Exist_a_Countable_Number_of_Computable_Numbers | [
"Computable Numbers",
"Countably Infinite Sets"
] | [
"Definition:Computable/Number",
"Definition:Countably Infinite/Set"
] | [
"Definition:Set",
"Definition:Computable/Number",
"Rational Number is Computable",
"Definition:Countably Infinite/Set",
"Definition:Algorithm",
"Definition:Computer Program",
"Definition:Set",
"Definition:Computer Program",
"Definition:Computable/Number",
"Definition:Subset",
"Definition:Set",
... |
proofwiki-22830 | Almost All Real Numbers are Not Computable | Almost all real numbers are not computable. | Let $S$ be the set of computable numbers.
From There Exist a Countable Number of Computable Numbers, $S$ is countably infinite.
From Real Numbers are Uncountable, the set of real numbers $\R$ is uncountable.
The result follows by definition of almost all.
{{qed}} | [[Definition:Almost All/Set Theory/Uncountable|Almost all]] [[Definition:Real Number|real numbers]] are not [[Definition:Computable Number|computable]]. | Let $S$ be the [[Definition:Set|set]] of [[Definition:Computable Number|computable numbers]].
From [[There Exist a Countable Number of Computable Numbers]], $S$ is [[Definition:Countably Infinite Set|countably infinite]].
From [[Real Numbers are Uncountable]], the [[Definition:Set|set]] of [[Definition:Real Number|re... | Almost All Real Numbers are Not Computable | https://proofwiki.org/wiki/Almost_All_Real_Numbers_are_Not_Computable | https://proofwiki.org/wiki/Almost_All_Real_Numbers_are_Not_Computable | [
"Computable Numbers",
"Real Numbers"
] | [
"Definition:Almost All/Set Theory/Uncountable",
"Definition:Real Number",
"Definition:Computable/Number"
] | [
"Definition:Set",
"Definition:Computable/Number",
"There Exist a Countable Number of Computable Numbers",
"Definition:Countably Infinite/Set",
"Real Numbers are Uncountably Infinite",
"Definition:Set",
"Definition:Real Number",
"Definition:Uncountable/Set",
"Definition:Almost All/Set Theory/Uncounta... |
proofwiki-22831 | Product Law of Independent Events | Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $A, B \in \Sigma$ be events of $\EE$ such that $\map \Pr A > 0$ and $\map \Pr B > 0$.
Let $A$ and $B$ be independent of each other.
Then:
:$\map \Pr {A \cap B} = \map \Pr A \map \Pr B$ | {{begin-eqn}}
{{eqn | l = \condprob A B
| r = \map \Pr A
| c = {{Defof|Independent Events|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}
| r = \map \Pr A
| c = {{Defof|Conditional Probability}}
}}
{{eqn | ll= \leadstoandfrom
| l = \map \Pr... | Let $\EE$ be an [[Definition:Experiment|experiment]] with [[Definition:Probability Space|probability space]] $\struct {\Omega, \Sigma, \Pr}$.
Let $A, B \in \Sigma$ be [[Definition:Event|events]] of $\EE$ such that $\map \Pr A > 0$ and $\map \Pr B > 0$.
Let $A$ and $B$ be [[Definition:Independent Events|independent]] ... | {{begin-eqn}}
{{eqn | l = \condprob A B
| r = \map \Pr A
| c = {{Defof|Independent Events|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}
| r = \map \Pr A
| c = {{Defof|Conditional Probability}}
}}
{{eqn | ll= \leadstoandfrom
| l = \map \Pr... | Product Law of Independent Events | https://proofwiki.org/wiki/Product_Law_of_Independent_Events | https://proofwiki.org/wiki/Product_Law_of_Independent_Events | [
"Independent Events",
"Named Theorems"
] | [
"Definition:Experiment",
"Definition:Probability Space",
"Definition:Event",
"Definition:Independent Events"
] | [] |
proofwiki-22832 | Equation of Conic Section/Cartesian Form/Circle | Let $a = b$ and $h = 0$.
Then $\CC$ is a circle. | When $a = b$ and $h = 0$, $(1)$ becomes:
:$a \paren {x^2 + y^2} + 2 g x + 2 f y + c = 0$
which is in the form specified in formulation $2$ of Equation of Circle.
{{qed}} | Let $a = b$ and $h = 0$.
Then $\CC$ is a [[Definition:Circle|circle]]. | When $a = b$ and $h = 0$, $(1)$ becomes:
:$a \paren {x^2 + y^2} + 2 g x + 2 f y + c = 0$
which is in the form specified in [[Equation of Circle/Cartesian/Formulation 2|formulation $2$ of Equation of Circle]].
{{qed}} | Equation of Conic Section/Cartesian Form/Circle | https://proofwiki.org/wiki/Equation_of_Conic_Section/Cartesian_Form/Circle | https://proofwiki.org/wiki/Equation_of_Conic_Section/Cartesian_Form/Circle | [
"Circles",
"Equation of Conic Section"
] | [
"Definition:Circle"
] | [
"Equation of Circle/Cartesian/Formulation 2"
] |
proofwiki-22833 | Moment of Inertia of Uniform Rectangular Lamina about Edge | Let $\LL$ be a uniform lamina of mass $M$ in the shape of a rectangle whose sides are of length $2 a$ and $2 b$.
Let $\AA$ be the straight line passing through the side of $\LL$ of length $2 b$.
Then the moment of inertia $\II$ of $\LL$ about $\AA$ is given by:
:$\II = \dfrac {4 M a^2} 3$ | Let $\AA'$ be the straight line:
:through the centroid of $\LL$
:in the plane of $\LL$
:perpendicular to the side of $\LL$ of length $2 a$.
Let $\II'$ be the moment of inertia of $\RR$ about $\AA'$.
We note that:
:$\AA'$ is parallel to $\AA$
:$\AA$ and $\AA'$ are at a distance $a$ apart.
Then:
{{begin-eqn}}
{{eqn | l =... | Let $\LL$ be a [[Definition:Uniform Lamina|uniform lamina]] of [[Definition:Mass|mass]] $M$ in the shape of a [[Definition:Rectangle|rectangle]] whose [[Definition:Side of Polygon|sides]] are of [[Definition:Length (Linear Measure)|length]] $2 a$ and $2 b$.
Let $\AA$ be the [[Definition:Straight Line|straight line]] p... | Let $\AA'$ be the [[Definition:Straight Line|straight line]]:
:through the [[Definition:Centroid of Surface|centroid]] of $\LL$
:in the [[Definition:Plane|plane]] of $\LL$
:[[Definition:Line Perpendicular to Plane|perpendicular]] to the [[Definition:Side of Polygon|side]] of $\LL$ of [[Definition:Length (Linear Measure... | Moment of Inertia of Uniform Rectangular Lamina about Edge | https://proofwiki.org/wiki/Moment_of_Inertia_of_Uniform_Rectangular_Lamina_about_Edge | https://proofwiki.org/wiki/Moment_of_Inertia_of_Uniform_Rectangular_Lamina_about_Edge | [
"Moments of Inertia",
"Uniform Laminae",
"Rectangles"
] | [
"Definition:Lamina/Uniform",
"Definition:Mass",
"Definition:Quadrilateral/Rectangle",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Line/Straight Line",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Moment of Inertia"
] | [
"Definition:Line/Straight Line",
"Definition:Centroid/Surface",
"Definition:Plane Surface",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Moment of Inertia",
"Definition:Parallel (Geometry)/Lines",
"Definition:Distance betwee... |
proofwiki-22834 | Moment of Inertia of Uniform Hoop about Diameter | Let $\HH$ be a hoop of uniform density of radius $r$ and mass $M$.
Let $\DD$ be a straight line through a diameter of $\HH$.
Then the moment of inertia $\II$ of $\HH$ about $\DD$ is given by:
:$\II = \dfrac {M r^2} 2$ | Let $\AA$ denote the axis of $\HH$.
Let $\DD_1$ and $\DD_2$ be two distinct diameters of $\HH$.
Let the moments of inertia of $\HH$ about $\DD_1$ and $\DD_2$ be $\II_1$ and $\II_2$.
By symmetry, $\II_1 = \II_2$ and both are equal to $\II$.
Let $\II_a$ be the moment of inertia of $\HH$ about $\AA$.
We note that $\AA$ is... | Let $\HH$ be a [[Definition:Hoop|hoop]] of [[Definition:Uniform Density|uniform density]] of [[Definition:Radius of Circle|radius]] $r$ and [[Definition:Mass|mass]] $M$.
Let $\DD$ be a [[Definition:Straight Line|straight line]] through a [[Definition:Diameter of Circle|diameter]] of $\HH$.
Then the [[Definition:Mome... | Let $\AA$ denote the [[Definition:Axis of Hoop|axis]] of $\HH$.
Let $\DD_1$ and $\DD_2$ be two [[Definition:Distinct|distinct]] [[Definition:Diameter of Circle|diameters]] of $\HH$.
Let the [[Definition:Moment of Inertia|moments of inertia]] of $\HH$ about $\DD_1$ and $\DD_2$ be $\II_1$ and $\II_2$.
By [[Definition... | Moment of Inertia of Uniform Hoop about Diameter | https://proofwiki.org/wiki/Moment_of_Inertia_of_Uniform_Hoop_about_Diameter | https://proofwiki.org/wiki/Moment_of_Inertia_of_Uniform_Hoop_about_Diameter | [
"Moments of Inertia",
"Hoops"
] | [
"Definition:Hoop",
"Definition:Uniform Density",
"Definition:Circle/Radius",
"Definition:Mass",
"Definition:Line/Straight Line",
"Definition:Circle/Diameter",
"Definition:Moment of Inertia"
] | [
"Definition:Hoop/Axis",
"Definition:Distinct",
"Definition:Circle/Diameter",
"Definition:Moment of Inertia",
"Definition:Symmetry",
"Definition:Moment of Inertia",
"Definition:Right Angle/Perpendicular",
"Moment of Inertia of Uniform Hoop about Axis",
"Perpendicular Axes Theorem",
"Moment of Inert... |
proofwiki-22835 | Cardinality of Conjugacy Class is Divisor of Order of Group | Let $G$ be a group.
For $x \in G$, let $\conjclass x$ denote the conjugacy class of $x$.
Then the cardinality of $\conjclass x$ is a divisor of the order of $G$. | {{ProofWanted|Follows obviously enough from Size of Conjugacy Class is Index of Normalizer}} | Let $G$ be a [[Definition:Group|group]].
For $x \in G$, let $\conjclass x$ denote the [[Definition:Conjugacy Class|conjugacy class]] of $x$.
Then the [[Definition:Cardinality|cardinality]] of $\conjclass x$ is a [[Definition:Divisor of Integer|divisor]] of the [[Definition:Order of Group|order]] of $G$. | {{ProofWanted|Follows obviously enough from [[Size of Conjugacy Class is Index of Normalizer]]}} | Cardinality of Conjugacy Class is Divisor of Order of Group | https://proofwiki.org/wiki/Cardinality_of_Conjugacy_Class_is_Divisor_of_Order_of_Group | https://proofwiki.org/wiki/Cardinality_of_Conjugacy_Class_is_Divisor_of_Order_of_Group | [
"Conjugacy Classes"
] | [
"Definition:Group",
"Definition:Conjugacy Class",
"Definition:Cardinality",
"Definition:Divisor (Algebra)/Integer",
"Definition:Order of Structure"
] | [
"Size of Conjugacy Class is Index of Normalizer"
] |
proofwiki-22836 | Product of Conjugate Quadratic Irrationals | Let $a + \sqrt b$ and $a - \sqrt b$ be conjugate quadratic irrationals.
Then their product is rational:
:$\paren {a + \sqrt b} \paren {a - \sqrt b} = a^2 - b$ | {{begin-eqn}}
{{eqn | l = \paren {a + \sqrt b} \paren {a - \sqrt b}
| r = a^2 - \paren {\sqrt b}^2
| c = Difference of Two Squares
}}
{{eqn | r = a^2 - b
| c =
}}
{{end-eqn}}
{{qed}} | Let $a + \sqrt b$ and $a - \sqrt b$ be [[Definition:Conjugate Quadratic Irrationals|conjugate quadratic irrationals]].
Then their [[Definition:Product (Algebra)|product]] is [[Definition:Rational Number|rational]]:
:$\paren {a + \sqrt b} \paren {a - \sqrt b} = a^2 - b$ | {{begin-eqn}}
{{eqn | l = \paren {a + \sqrt b} \paren {a - \sqrt b}
| r = a^2 - \paren {\sqrt b}^2
| c = [[Difference of Two Squares]]
}}
{{eqn | r = a^2 - b
| c =
}}
{{end-eqn}}
{{qed}} | Product of Conjugate Quadratic Irrationals | https://proofwiki.org/wiki/Product_of_Conjugate_Quadratic_Irrationals | https://proofwiki.org/wiki/Product_of_Conjugate_Quadratic_Irrationals | [
"Conjugates of Quadratic Irrationals"
] | [
"Definition:Conjugate of Quadratic Irrational",
"Definition:Multiplication/Product",
"Definition:Rational Number"
] | [
"Difference of Two Squares"
] |
proofwiki-22837 | Sum of Conjugate Quadratic Irrationals | Let $a + \sqrt b$ and $a - \sqrt b$ be conjugate quadratic irrationals.
Then their sum is rational:
:$\paren {a + \sqrt b} + \paren {a - \sqrt b} = 2 a$ | Simple algebra.
{{qed}} | Let $a + \sqrt b$ and $a - \sqrt b$ be [[Definition:Conjugate Quadratic Irrationals|conjugate quadratic irrationals]].
Then their [[Definition:Sum (Addition)|sum]] is [[Definition:Rational Number|rational]]:
:$\paren {a + \sqrt b} + \paren {a - \sqrt b} = 2 a$ | Simple [[Definition:Algebra (Mathematical Branch)|algebra]].
{{qed}} | Sum of Conjugate Quadratic Irrationals | https://proofwiki.org/wiki/Sum_of_Conjugate_Quadratic_Irrationals | https://proofwiki.org/wiki/Sum_of_Conjugate_Quadratic_Irrationals | [
"Conjugates of Quadratic Irrationals"
] | [
"Definition:Conjugate of Quadratic Irrational",
"Definition:Addition/Sum",
"Definition:Rational Number"
] | [
"Definition:Algebra (Mathematical Branch)"
] |
proofwiki-22838 | Epimorphism in Isomorphic Category | Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $F: \mathbf C \to \mathbf D$ be an isomorphism of categories.
Let $f$ be a morphism of $\mathbf C$.
Then $f$ is an epimorphism in $\mathbf C$ {{iff}} $Ff$ is an epimorphism in $\mathbf D$. | Let $G: \mathbf D \to \mathbf C$ denote the inverse of $F$.
By definition of inverse Isomorphism of categories:
:$G F: \mathbf C \to \mathbf C$ is the identity functor $\operatorname{id}_{\mathbf C}$
:$F G: \mathbf D \to \mathbf D$ is the identity functor $\operatorname{id}_{\mathbf D}$ | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Metacategory|metacategories]].
Let $F: \mathbf C \to \mathbf D$ be an [[Definition:Isomorphism of Categories|isomorphism of categories]].
Let $f$ be a [[Definition:Morphism (Category Theory)|morphism]] of $\mathbf C$.
Then $f$ is an [[Definition:Epimorphism (Category ... | Let $G: \mathbf D \to \mathbf C$ denote the [[Definition:Inverse Isomorphism of Categories|inverse]] of $F$.
By definition of [[Definition:Inverse Isomorphism of Categories|inverse Isomorphism of categories]]:
:$G F: \mathbf C \to \mathbf C$ is the [[Definition:Identity Functor|identity functor]] $\operatorname{id}_{\... | Epimorphism in Isomorphic Category | https://proofwiki.org/wiki/Epimorphism_in_Isomorphic_Category | https://proofwiki.org/wiki/Epimorphism_in_Isomorphic_Category | [
"Epimorphisms (Category Theory)",
"Isomorphisms of Categories"
] | [
"Definition:Metacategory",
"Definition:Isomorphism of Categories",
"Definition:Morphism",
"Definition:Epimorphism (Category Theory)",
"Definition:Epimorphism (Category Theory)"
] | [
"Definition:Isomorphism of Categories/Inverse",
"Definition:Isomorphism of Categories/Inverse",
"Definition:Identity Functor",
"Definition:Identity Functor"
] |
proofwiki-22839 | Monomorphism in Isomorphic Category | Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $F: \mathbf C \to \mathbf D$ be an isomorphism of categories.
Let $f$ be a morphism of $\mathbf C$.
Then $f$ is a monomorphism in $\mathbf C$ {{iff}} $Ff$ is a monomorphism in $\mathbf D$. | Let $G: \mathbf D \to \mathbf C$ denote the inverse of $F$.
By definition of inverse Isomorphism of categories:
:$G F: \mathbf C \to \mathbf C$ is the identity functor $\operatorname{id}_{\mathbf C}$
:$F G: \mathbf D \to \mathbf D$ is the identity functor $\operatorname{id}_{\mathbf D}$ | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Metacategory|metacategories]].
Let $F: \mathbf C \to \mathbf D$ be an [[Definition:Isomorphism of Categories|isomorphism of categories]].
Let $f$ be a [[Definition:Morphism (Category Theory)|morphism]] of $\mathbf C$.
Then $f$ is a [[Definition:Monomorphism (Category ... | Let $G: \mathbf D \to \mathbf C$ denote the [[Definition:Inverse Isomorphism of Categories|inverse]] of $F$.
By definition of [[Definition:Inverse Isomorphism of Categories|inverse Isomorphism of categories]]:
:$G F: \mathbf C \to \mathbf C$ is the [[Definition:Identity Functor|identity functor]] $\operatorname{id}_{\... | Monomorphism in Isomorphic Category | https://proofwiki.org/wiki/Monomorphism_in_Isomorphic_Category | https://proofwiki.org/wiki/Monomorphism_in_Isomorphic_Category | [
"Monomorphisms (Category Theory)",
"Isomorphisms of Categories"
] | [
"Definition:Metacategory",
"Definition:Isomorphism of Categories",
"Definition:Morphism",
"Definition:Monomorphism (Category Theory)",
"Definition:Monomorphism (Category Theory)"
] | [
"Definition:Isomorphism of Categories/Inverse",
"Definition:Isomorphism of Categories/Inverse",
"Definition:Identity Functor",
"Definition:Identity Functor"
] |
proofwiki-22840 | Connected Component is not necessarily Open | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq T$ be a connected component of $T$.
Then it is not necessarily the case that $H$ is open. | Let $T = \struct {\Q, \tau_d}$ be the rational number space formed by the rational numbers $\Q$ under the usual (Euclidean) topology $\tau_d$.
Let $x \in \Q$.
From Components of Rational Number Space are Singletons, $\set x$ is a connected component of $T$.
{{finish|Need to establish that $\set x$ is not an open set of... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq T$ be a [[Definition:Component (Topology)|connected component]] of $T$.
Then it is not necessarily the case that $H$ is [[Definition:Open Set (Topology)|open]]. | Let $T = \struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] formed by the [[Definition:Rational Number|rational numbers]] $\Q$ under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]] $\tau_d$.
Let $x \in \Q$.
From [[Components of Rational Number ... | Connected Component is not necessarily Open | https://proofwiki.org/wiki/Connected_Component_is_not_necessarily_Open | https://proofwiki.org/wiki/Connected_Component_is_not_necessarily_Open | [
"Components (Topology)",
"Open Sets"
] | [
"Definition:Topological Space",
"Definition:Component (Topology)",
"Definition:Open Set/Topology"
] | [
"Definition:Rational Number Space",
"Definition:Rational Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Components of Rational Number Space are Singletons",
"Definition:Component (Topology)"
] |
proofwiki-22841 | Connected Space is not necessarily Path-Connected | Let $T$ be a topological space which is connected.
Then it is not necessarily the case that $T$ is path-connected. | Let $T$ be the closed topologist's sine curve.
From Closed Topologist's Sine Curve is not Path-Connected, $T$ is path-connected.
From Closed Topologist's Sine Curve is Connected, $T$ is connected.
Hence the result.
{{qed}} | Let $T$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Connected Topological Space|connected]].
Then it is not necessarily the case that $T$ is [[Definition:Path-Connected Space|path-connected]]. | Let $T$ be the [[Definition:Closed Topologist's Sine Curve|closed topologist's sine curve]].
From [[Closed Topologist's Sine Curve is not Path-Connected]], $T$ is [[Definition:Path-Connected Space|path-connected]].
From [[Closed Topologist's Sine Curve is Connected]], $T$ is [[Definition:Connected Topological Space|c... | Connected Space is not necessarily Path-Connected | https://proofwiki.org/wiki/Connected_Space_is_not_necessarily_Path-Connected | https://proofwiki.org/wiki/Connected_Space_is_not_necessarily_Path-Connected | [
"Path-Connected Spaces",
"Connected Topological Spaces",
"Sequence of Implications of Connectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Connected Topological Space",
"Definition:Path-Connected/Topological Space"
] | [
"Definition:Closed Topologist's Sine Curve",
"Closed Topologist's Sine Curve is not Path-Connected",
"Definition:Path-Connected/Topological Space",
"Closed Topologist's Sine Curve is Connected",
"Definition:Connected Topological Space"
] |
proofwiki-22842 | Isomorphism in Isomorphic Category | Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $F: \mathbf C \to \mathbf D$ be an isomorphism of categories.
Let $f$ be a morphism of $\mathbf C$.
Then $f$ is an isomorphism in $\mathbf C$ {{iff}} $Ff$ is an isomorphism in $\mathbf D$. | Let $G: \mathbf D \to \mathbf C$ denote the inverse of $F$.
By definition of inverse Isomorphism of categories:
:$G F: \mathbf C \to \mathbf C$ is the identity functor $\operatorname{id}_{\mathbf C}$
:$F G: \mathbf D \to \mathbf D$ is the identity functor $\operatorname{id}_{\mathbf D}$ | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Metacategory|metacategories]].
Let $F: \mathbf C \to \mathbf D$ be an [[Definition:Isomorphism of Categories|isomorphism of categories]].
Let $f$ be a [[Definition:Morphism (Category Theory)|morphism]] of $\mathbf C$.
Then $f$ is an [[Definition:Isomorphism (Category ... | Let $G: \mathbf D \to \mathbf C$ denote the [[Definition:Inverse Isomorphism of Categories|inverse]] of $F$.
By definition of [[Definition:Inverse Isomorphism of Categories|inverse Isomorphism of categories]]:
:$G F: \mathbf C \to \mathbf C$ is the [[Definition:Identity Functor|identity functor]] $\operatorname{id}_{\... | Isomorphism in Isomorphic Category | https://proofwiki.org/wiki/Isomorphism_in_Isomorphic_Category | https://proofwiki.org/wiki/Isomorphism_in_Isomorphic_Category | [
"Isomorphisms (Category Theory)",
"Isomorphisms of Categories"
] | [
"Definition:Metacategory",
"Definition:Isomorphism of Categories",
"Definition:Morphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)"
] | [
"Definition:Isomorphism of Categories/Inverse",
"Definition:Isomorphism of Categories/Inverse",
"Definition:Identity Functor",
"Definition:Identity Functor"
] |
proofwiki-22843 | Ultrafilter is Principal iff Contains a Finite Set | Let $\UU \subseteq \powerset S$ be a ultrafilter on $S$.
Then:
:$\UU$ is principal
{{iff}}:
:there is a finite $A \in \UU$ | === Necessary Condition ===
By Filter is Principal Ultrafilter iff Contains Singleton, there is some $x \in S$ such that:
:$\set x \in \UU$
Putting $A = \set x$, the result follows from Singleton is Finite.
{{qed|lemma}} | Let $\UU \subseteq \powerset S$ be a [[Definition:Ultrafilter on Set|ultrafilter]] on $S$.
Then:
:$\UU$ is [[Definition:Principal Ultrafilter|principal]]
{{iff}}:
:there is a [[Definition:Finite Set|finite]] $A \in \UU$ | === Necessary Condition ===
By [[Filter is Principal Ultrafilter iff Contains Singleton]], there is some $x \in S$ such that:
:$\set x \in \UU$
Putting $A = \set x$, the result follows from [[Singleton is Finite]].
{{qed|lemma}} | Ultrafilter is Principal iff Contains a Finite Set | https://proofwiki.org/wiki/Ultrafilter_is_Principal_iff_Contains_a_Finite_Set | https://proofwiki.org/wiki/Ultrafilter_is_Principal_iff_Contains_a_Finite_Set | [
"Ultrafilters on Sets",
"Principal Ultrafilters"
] | [
"Definition:Ultrafilter on Set",
"Definition:Principal Ultrafilter",
"Definition:Finite Set"
] | [
"Filter is Principal Ultrafilter iff Contains Singleton",
"Singleton is Finite",
"Filter is Principal Ultrafilter iff Contains Singleton"
] |
proofwiki-22844 | Filter is Principal Ultrafilter iff Contains Singleton | Let $\FF \subseteq \powerset S$ be a filter on $S$.
Then:
:$\FF$ is a principal ultrafilter on $S$
{{iff}}:
:$\set x \in \FF$ for some $x \in S$ | === Necessary Condition ===
Suppose $\FF$ is a principal ultrafilter on $S$.
By definition, it has a cluster point $x \in S$.
By Principal Ultrafilter is All Sets Containing Cluster Point, we have in particular that:
:$\set x \in S$
{{qed|lemma}} | Let $\FF \subseteq \powerset S$ be a [[Definition:Filter on Set|filter]] on $S$.
Then:
:$\FF$ is a [[Definition:Principal Ultrafilter|principal ultrafilter]] on $S$
{{iff}}:
:$\set x \in \FF$ for some $x \in S$ | === Necessary Condition ===
Suppose $\FF$ is a [[Definition:Principal Ultrafilter|principal ultrafilter]] on $S$.
By definition, it has a [[Definition:Cluster Point of Filter|cluster point]] $x \in S$.
By [[Principal Ultrafilter is All Sets Containing Cluster Point]], we have in particular that:
:$\set x \in S$
{{qe... | Filter is Principal Ultrafilter iff Contains Singleton | https://proofwiki.org/wiki/Filter_is_Principal_Ultrafilter_iff_Contains_Singleton | https://proofwiki.org/wiki/Filter_is_Principal_Ultrafilter_iff_Contains_Singleton | [
"Principal Ultrafilters",
"Filters on Sets"
] | [
"Definition:Filter on Set",
"Definition:Principal Ultrafilter"
] | [
"Definition:Principal Ultrafilter",
"Definition:Cluster Point of Filter",
"Principal Ultrafilter is All Sets Containing Cluster Point",
"Definition:Cluster Point of Filter",
"Definition:Principal Ultrafilter"
] |
proofwiki-22845 | Filter Contains no Disjoint Sets | Let $\FF \subseteq \powerset S$ be a filter on $S$.
Let $A, B \subseteq S$ be disjoint.
Then, at most one of $A$ or $B$ is an element of $\FF$. | {{AimForCont}} both $A \in \FF$ and $B \in \FF$.
Then, by filter axiom $\text F 3$:
:$A \cap B \in \FF$
But:
:$A \cap B = \O$
by definition of disjoint, which contradicts filter axiom $\text F 2$.
The result follows by Proof by Contradiction.
{{qed}}
Category:Filter Theory
8k8gcblt5ciuoa8brbmedi1jfx9256m | Let $\FF \subseteq \powerset S$ be a [[Definition:Filter on Set|filter]] on $S$.
Let $A, B \subseteq S$ be [[Definition:Disjoint Sets|disjoint]].
Then, at most one of $A$ or $B$ is an [[Definition:Element|element]] of $\FF$. | {{AimForCont}} both $A \in \FF$ and $B \in \FF$.
Then, by [[Axiom:Filter on Set Axioms|filter axiom $\text F 3$]]:
:$A \cap B \in \FF$
But:
:$A \cap B = \O$
by definition of [[Definition:Disjoint Sets|disjoint]], which contradicts [[Axiom:Filter on Set Axioms|filter axiom $\text F 2$]].
The result follows by [[Proof... | Filter Contains no Disjoint Sets | https://proofwiki.org/wiki/Filter_Contains_no_Disjoint_Sets | https://proofwiki.org/wiki/Filter_Contains_no_Disjoint_Sets | [
"Filter Theory"
] | [
"Definition:Filter on Set",
"Definition:Disjoint Sets",
"Definition:Element"
] | [
"Axiom:Filter on Set Axioms",
"Definition:Disjoint Sets",
"Axiom:Filter on Set Axioms",
"Proof by Contradiction",
"Category:Filter Theory"
] |
proofwiki-22846 | Ultrafilter Contains Set or Complement | Let $\UU \subseteq \powerset S$ be a ultrafilter on $S$.
Let $A \subseteq S$.
Then, exactly one of $A$ and $\relcomp S A$ is an element of $\UU$. | At least one is an element by definition of an ultrafilter.
Moreover, at most one is by Filter Contains no Disjoint Sets, since:
:$A \cap \relcomp S A = \O$
The result follows.
{{qed}}
Category:Ultrafilters on Sets
nr1pvkot0ch2je8rzkjs3b0rx6mhrni | Let $\UU \subseteq \powerset S$ be a [[Definition:Ultrafilter on Set|ultrafilter]] on $S$.
Let $A \subseteq S$.
Then, exactly one of $A$ and $\relcomp S A$ is an [[Definition:Element|element]] of $\UU$. | At least one is an [[Definition:Element|element]] by definition of an [[Definition:Ultrafilter on Set/Definition 3|ultrafilter]].
Moreover, at most one is by [[Filter Contains no Disjoint Sets]], since:
:$A \cap \relcomp S A = \O$
The result follows.
{{qed}}
[[Category:Ultrafilters on Sets]]
nr1pvkot0ch2je8rzkjs3b0r... | Ultrafilter Contains Set or Complement | https://proofwiki.org/wiki/Ultrafilter_Contains_Set_or_Complement | https://proofwiki.org/wiki/Ultrafilter_Contains_Set_or_Complement | [
"Ultrafilters on Sets"
] | [
"Definition:Ultrafilter on Set",
"Definition:Element"
] | [
"Definition:Element",
"Definition:Ultrafilter on Set/Definition 3",
"Filter Contains no Disjoint Sets",
"Category:Ultrafilters on Sets"
] |
proofwiki-22847 | Ultrafilter is Nonprincipal iff Contains Fréchet Filter | Let $\UU \subseteq \powerset S$ be an ultrafilter on $S$.
Then:
:$\UU$ is nonprincipal
{{iff}}:
:$\FF \subseteq \UU$, where $\FF$ is the Fréchet filter on $S$. | By Ultrafilter Contains Set or Complement, we have that:
:$A \notin \UU \iff \relcomp S A \in \UU$
for each $A \subseteq S$.
Considering $A$ to range over all finite subsets of $S$, we find that:
:$A \notin \UU$ for every finite $A \subseteq S$
{{iff}}:
:$\relcomp S A \in \UU$ for every finite $A \subseteq S$
But the l... | Let $\UU \subseteq \powerset S$ be an [[Definition:Ultrafilter on Set|ultrafilter]] on $S$.
Then:
:$\UU$ is [[Definition:Nonprincipal Ultrafilter|nonprincipal]]
{{iff}}:
:$\FF \subseteq \UU$, where $\FF$ is the [[Definition:Fréchet Filter|Fréchet filter]] on $S$. | By [[Ultrafilter Contains Set or Complement]], we have that:
:$A \notin \UU \iff \relcomp S A \in \UU$
for each $A \subseteq S$.
Considering $A$ to range over all [[Definition:Finite Subset|finite subsets]] of $S$, we find that:
:$A \notin \UU$ for every [[Definition:Finite Set|finite]] $A \subseteq S$
{{iff}}:
:$\rel... | Ultrafilter is Nonprincipal iff Contains Fréchet Filter | https://proofwiki.org/wiki/Ultrafilter_is_Nonprincipal_iff_Contains_Fréchet_Filter | https://proofwiki.org/wiki/Ultrafilter_is_Nonprincipal_iff_Contains_Fréchet_Filter | [
"Fréchet Filters",
"Nonprincipal Ultrafilters",
"Ultrafilters on Sets"
] | [
"Definition:Ultrafilter on Set",
"Definition:Principal Ultrafilter/Nonprincipal",
"Definition:Fréchet Filter"
] | [
"Ultrafilter Contains Set or Complement",
"Definition:Finite Subset",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Cofinite Subset",
"Definition:Fréchet Filter",
"Definition:Principal Ultrafilter/Nonprincipal",
"Definition:Finite Set",
"Ultrafilter is Principal iff Contains a Finite... |
proofwiki-22848 | Dual of Dual Category is Isomorphic to Category | Let $\mathbf C$ be a metacategory.
Let $\mathbf C^{\text{op} }$ be the dual category of $\mathbf C$.
Let $\paren{\mathbf C^{\text{op} } }^{\text{op} }$ be the dual category of $\mathbf C^{\text{op} }$.
Then:
:$C$ is isomorphic to $\paren{\mathbf C^{\text{op} } }^{\text{op} }$
with isomorphisms:
:$F: C \to \paren{\mathb... | === $F$ is a Functor ===
Let $f, g$ be morphisms of $\mathbf C$ such that $g \circ f$ is defined.
Then:
{{begin-eqn}}
{{eqn | l = \map F {g \circ f}
| r = \paren{\paren{g \circ f}^{\text{op} } }^{\text{op} }
| c = Definition of $F$
}}
{{eqn | r = \paren{f^{\text{op} } \circ g^{\text{op} } }^{\text{op} }
... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $\mathbf C^{\text{op} }$ be the [[Definition:Dual Category|dual category]] of $\mathbf C$.
Let $\paren{\mathbf C^{\text{op} } }^{\text{op} }$ be the [[Definition:Dual Category|dual category]] of $\mathbf C^{\text{op} }$.
Then:
:$C$ is [[Definition:I... | === $F$ is a Functor ===
Let $f, g$ be [[Definition:Morphism (Category Theory)|morphisms]] of $\mathbf C$ such that $g \circ f$ is defined.
Then:
{{begin-eqn}}
{{eqn | l = \map F {g \circ f}
| r = \paren{\paren{g \circ f}^{\text{op} } }^{\text{op} }
| c = Definition of $F$
}}
{{eqn | r = \paren{f^{\text{op... | Dual of Dual Category is Isomorphic to Category | https://proofwiki.org/wiki/Dual_of_Dual_Category_is_Isomorphic_to_Category | https://proofwiki.org/wiki/Dual_of_Dual_Category_is_Isomorphic_to_Category | [
"Dual Categories",
"Isomorphisms of Categories"
] | [
"Definition:Metacategory",
"Definition:Dual Category",
"Definition:Dual Category",
"Definition:Isomorphism of Categories/Isomorphic Categories",
"Definition:Isomorphism of Categories",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Theory)",
"Definition:Mo... | [
"Definition:Morphism",
"Definition:Dual Category",
"Definition:Dual Category",
"Definition:Object",
"Definition:Dual Category",
"Definition:Dual Category",
"Definition:Functor/Covariant",
"Definition:Morphism",
"Definition:Dual Category",
"Definition:Dual Category",
"Definition:Object",
"Defin... |
proofwiki-22849 | Normalized Block Basic Sequence of Schauder Basis in Space of Zero-Limit Sequences is Isometrically Equivalent to Schauder Basis | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
Let $\sequence {e_n}_{n \mathop \in \N}$ be the Schauder basis given by Space of Zero-Limit Sequences admits Schauder Basis.
Let $\sequence {y_k}_{k \mathop \in \N}$ be a block basic sequence of... | Let $\sequence {r_j}_{j \mathop \in \N}$ be a strictly increasing sequence in $\N$ and $\sequence {a_j}_{j \mathop \in \N}$ be a sequence in $\GF$ such that:
:$\ds y_k = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} a_j e_j$
where $r_0 = 0$.
By assumption, we have:
:$\ds 1 = \norm {y_k}_\infty = \sup_{r_{k - 1} + 1 \le j \le ... | 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 $\sequence {e_n}_{n \mathop \in \N}$ be the [[Definition:Schauder Basis|Schauder basis]] given by [[Space of Zero-Limit Sequences admits Schaud... | Let $\sequence {r_j}_{j \mathop \in \N}$ be a [[Definition:Strictly Increasing Sequence|strictly increasing sequence]] in $\N$ and $\sequence {a_j}_{j \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\GF$ such that:
:$\ds y_k = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} a_j e_j$
where $r_0 = 0$.
By assumption, w... | Normalized Block Basic Sequence of Schauder Basis in Space of Zero-Limit Sequences is Isometrically Equivalent to Schauder Basis | https://proofwiki.org/wiki/Normalized_Block_Basic_Sequence_of_Schauder_Basis_in_Space_of_Zero-Limit_Sequences_is_Isometrically_Equivalent_to_Schauder_Basis | https://proofwiki.org/wiki/Normalized_Block_Basic_Sequence_of_Schauder_Basis_in_Space_of_Zero-Limit_Sequences_is_Isometrically_Equivalent_to_Schauder_Basis | [
"Basic Sequences",
"Block Basic Sequences",
"Block Basic Sequences",
"Space of Zero-Limit Sequences"
] | [
"Definition:Space of Zero-Limit Sequences",
"Definition:Schauder Basis",
"Space of Zero-Limit Sequences admits Schauder Basis",
"Definition:Block Basic Sequence",
"Definition:Isometric Equivalence of Basic Sequences"
] | [
"Definition:Strictly Increasing/Sequence",
"Definition:Sequence",
"Definition:Space of Almost-Zero Sequences",
"Characterization of Isometric Equivalence of Basic Sequences",
"Characterization of Isometric Equivalence of Basic Sequences",
"Definition:Isometric Equivalence of Basic Sequences"
] |
proofwiki-22850 | Space of Almost-Zero Sequences is Everywhere Dense in P-Sequence Space | Let $\GF \in \set {\R, \C}$.
Let $p \in \hointr 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the $p$-sequence space.
Let $\map {c_{00} } \GF$ be the space of almost-zero sequences.
Then $\map {c_{00} } \GF$ is everywhere dense in $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$. | Define $\sequence {e_n}_{n \mathop \in \N}$ by:
:$\ds \map {e_n} k = \begin{cases}1 & n = k \\ 0 & \text{otherwise}\end{cases} \in \map {c_{00} } \GF$
From P-Sequence Space admits Schauder Basis, $\sequence {e_n}_{n \mathop \in \N}$ is a Schauder basis for $\map {\ell_p} \GF$.
Let $\sequence {e_n^\ast}_{n \mathop \in \... | Let $\GF \in \set {\R, \C}$.
Let $p \in \hointr 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the [[Definition:P-Sequence Space|$p$-sequence space]].
Let $\map {c_{00} } \GF$ be the [[Definition:Space of Almost-Zero Sequences|space of almost-zero sequences]].
Then $\map {c_{00} } \GF$ is ... | Define $\sequence {e_n}_{n \mathop \in \N}$ by:
:$\ds \map {e_n} k = \begin{cases}1 & n = k \\ 0 & \text{otherwise}\end{cases} \in \map {c_{00} } \GF$
From [[P-Sequence Space admits Schauder Basis]], $\sequence {e_n}_{n \mathop \in \N}$ is a [[Definition:Schauder Basis|Schauder basis]] for $\map {\ell_p} \GF$.
Let $\... | Space of Almost-Zero Sequences is Everywhere Dense in P-Sequence Space | https://proofwiki.org/wiki/Space_of_Almost-Zero_Sequences_is_Everywhere_Dense_in_P-Sequence_Space | https://proofwiki.org/wiki/Space_of_Almost-Zero_Sequences_is_Everywhere_Dense_in_P-Sequence_Space | [
"Space of Almost-Zero Sequences",
"P-Sequence Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:Space of Almost-Zero Sequences",
"Definition:Everywhere Dense"
] | [
"P-Sequence Space admits Schauder Basis",
"Definition:Schauder Basis",
"Definition:Coordinate Functionals Associated with Schauder Basis",
"Definition:Coordinate Functionals Associated with Schauder Basis",
"Space of Almost-Zero Sequences forms Vector Space",
"Definition:Closure (Topology)",
"Definition... |
proofwiki-22851 | Conditions for Constant Speed | Let $B$ be a body moving at constant speed.
Then the velocity of $B$ is not necessarily constant. | Let $\mathbf v$ be the velocity of body $B$.
The speed of $B$ is given by the magnitude of the velocity:
:$v = \size {\mathbf v}$
We have that the speed is constant:
:$\ds \frac {\d v} {\d t} = 0$
As the speed is constant, then so is the square of the speed:
{{begin-eqn}}
{{eqn | l = \frac {\d v^2} {\d t}
| r = 2... | Let $B$ be a [[Definition:Body|body]] [[Definition:Motion|moving]] at [[Definition:Constant Speed|constant speed]].
Then the [[Definition:Velocity|velocity]] of $B$ is not necessarily [[Definition:Constant|constant]]. | Let $\mathbf v$ be the [[Definition:Velocity|velocity]] of [[Definition:Body|body]] $B$.
The [[Definition:Speed|speed]] of $B$ is given by the [[Definition:Magnitude|magnitude]] of the [[Definition:Velocity|velocity]]:
:$v = \size {\mathbf v}$
We have that the [[Definition:Speed|speed]] is [[Definition:Constant|const... | Conditions for Constant Speed | https://proofwiki.org/wiki/Conditions_for_Constant_Speed | https://proofwiki.org/wiki/Conditions_for_Constant_Speed | [
"Constant Speed"
] | [
"Definition:Body",
"Definition:Motion",
"Definition:Constant Speed",
"Definition:Velocity",
"Definition:Constant"
] | [
"Definition:Velocity",
"Definition:Body",
"Definition:Speed",
"Definition:Magnitude",
"Definition:Velocity",
"Definition:Speed",
"Definition:Constant",
"Definition:Speed",
"Definition:Constant",
"Definition:Square/Function",
"Definition:Speed",
"Derivative of Composite Function",
"Power Rule... |
proofwiki-22852 | Equivalence of Definitions of Boolean Lattice/Definition 3 implies Definition 1 | Let $\BB = \struct {S, \vee, \wedge, \preceq, \bot, \top}$ be a bounded lattice.
Let $\neg$ be a unary operation on $S$ such that:
:$\paren 1 \quad \forall a, b \in S: a \preceq \neg b \iff a \wedge b = \bot$
:$\paren 2 \quad \forall a \in S: \neg \neg a = a$
Then, $\BB$ is complemented and distributive. | Let us begin with some identities.
First:
:$\paren R \quad a \preceq \neg b \iff b \preceq \neg a$
since both are equivalent to $a \wedge b = \bot$ by $\paren 1$.
Second:
:$\paren M \quad a \preceq \neg \paren {\neg a \wedge b}$
which follows by $\paren R$ from:
:$\neg a \wedge b \preceq \neg a$
because Meet Precedes O... | Let $\BB = \struct {S, \vee, \wedge, \preceq, \bot, \top}$ be a [[Definition:Bounded Lattice|bounded lattice]].
Let $\neg$ be a [[Definition:Unary Operation|unary operation]] on $S$ such that:
:$\paren 1 \quad \forall a, b \in S: a \preceq \neg b \iff a \wedge b = \bot$
:$\paren 2 \quad \forall a \in S: \neg \neg a = ... | Let us begin with some identities.
First:
:$\paren R \quad a \preceq \neg b \iff b \preceq \neg a$
since both are [[Definition:Logical Equivalence|equivalent]] to $a \wedge b = \bot$ by $\paren 1$.
Second:
:$\paren M \quad a \preceq \neg \paren {\neg a \wedge b}$
which follows by $\paren R$ from:
:$\neg a \wedge b \p... | Equivalence of Definitions of Boolean Lattice/Definition 3 implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Boolean_Lattice/Definition_3_implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Boolean_Lattice/Definition_3_implies_Definition_1 | [] | [
"Definition:Bounded Lattice",
"Definition:Operation/Unary Operation",
"Definition:Complemented Lattice",
"Definition:Distributive Lattice"
] | [
"Definition:Logical Equivalence",
"Meet Precedes Operands",
"Meet Absorbs Join",
"Definition:Join (Order Theory)",
"Definition:Meet (Order Theory)",
"Axiom:Ordering Axioms",
"Join Succeeds Operands",
"Meet is Increasing",
"Meet Precedes Operands",
"Definition:Meet (Order Theory)",
"Axiom:Orderin... |
proofwiki-22853 | Closed Linear Span of Normalized Block Basic Sequence of Schauder Basis in P-Sequence Space is Range of Contractive Projection | Let $\GF \in \set {\R, \C}$.
Let $p \in \hointr 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the $p$-sequence space.
Let $\sequence {e_n}_{n \mathop \in \N}$ be the Schauder basis given by P-Sequence Space admits Schauder Basis.
Let $\sequence {y_k}_{k \mathop \in \N}$ be a block basic seque... | Let $\sequence {r_j}_{j \mathop \in \N}$ be a strictly increasing sequence in $\N$ and $\sequence {a_j}_{j \mathop \in \N}$ be a sequence in $\GF$ such that:
:$\ds y_k = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} a_j e_j$
where $r_0 = 0$.
We then have:
:$\ds 1 = \norm {y_k}_p = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} \cmod ... | Let $\GF \in \set {\R, \C}$.
Let $p \in \hointr 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the [[Definition:P-Sequence Space|$p$-sequence space]].
Let $\sequence {e_n}_{n \mathop \in \N}$ be the [[Definition:Schauder Basis|Schauder basis]] given by [[P-Sequence Space admits Schauder Bas... | Let $\sequence {r_j}_{j \mathop \in \N}$ be a [[Definition:Strictly Increasing Sequence|strictly increasing sequence]] in $\N$ and $\sequence {a_j}_{j \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\GF$ such that:
:$\ds y_k = \sum_{j \mathop = r_{k - 1} + 1}^{r_k} a_j e_j$
where $r_0 = 0$.
We then have:
:$... | Closed Linear Span of Normalized Block Basic Sequence of Schauder Basis in P-Sequence Space is Range of Contractive Projection | https://proofwiki.org/wiki/Closed_Linear_Span_of_Normalized_Block_Basic_Sequence_of_Schauder_Basis_in_P-Sequence_Space_is_Range_of_Contractive_Projection | https://proofwiki.org/wiki/Closed_Linear_Span_of_Normalized_Block_Basic_Sequence_of_Schauder_Basis_in_P-Sequence_Space_is_Range_of_Contractive_Projection | [
"Basic Sequences",
"Block Basic Sequences",
"Block Basic Sequences",
"P-Sequence Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:Schauder Basis",
"P-Sequence Space admits Schauder Basis",
"Definition:Block Basic Sequence",
"Definition:Closed Linear Span",
"Definition:Bounded Linear Transformation",
"Definition:Projection (Vector Spaces)",
"Definition:Topologically Complemented Subspace... | [
"Definition:Strictly Increasing/Sequence",
"Definition:Sequence",
"Block Basic Sequence is Basic Sequence",
"Definition:Schauder Basis",
"Definition:Coordinate Functionals Associated with Schauder Basis",
"Definition:Coordinate Functionals Associated with Schauder Basis",
"Condition for Sequence to be S... |
proofwiki-22854 | Basic Sequence Congruent to Complemented Basic Sequence is Complemented | 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 complemented basic sequence.
Let $\sequence {y_n}_{n \mathop \in \N}$ be a basic sequence in $Y$ that is congruent to $\sequence {x_n}... | Since $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ are congruent, there exists a linear isomorphism $T : X \to Y$ with $T x_n = y_n$ for each $n \in \N$.
Let $\sqbrk {x_n}_{n \mathop \in \N}$ and $\sqbrk {y_n}_{n \mathop \in \N}$ be the closed linear spans of $\sequence {x_n}_{n \mathop... | 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:Complemented Basic Sequence|complemented basic sequence]].
Let $\sequence {y_n}_{n \mathop... | Since $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ are [[Definition:Congruent Sequences|congruent]], there exists a [[Definition:Linear Isomorphism|linear isomorphism]] $T : X \to Y$ with $T x_n = y_n$ for each $n \in \N$.
Let $\sqbrk {x_n}_{n \mathop \in \N}$ and $\sqbrk {y_n}_{n \mat... | Basic Sequence Congruent to Complemented Basic Sequence is Complemented | https://proofwiki.org/wiki/Basic_Sequence_Congruent_to_Complemented_Basic_Sequence_is_Complemented | https://proofwiki.org/wiki/Basic_Sequence_Congruent_to_Complemented_Basic_Sequence_is_Complemented | [
"Basic Sequences"
] | [
"Definition:Banach Space",
"Definition:Complemented Basic Sequence",
"Definition:Basic Sequence",
"Definition:Congruent Sequences",
"Definition:Complemented Basic Sequence"
] | [
"Definition:Congruent Sequences",
"Definition:Linear Isomorphism",
"Definition:Closed Linear Span",
"Definition:Complemented Basic Sequence",
"Definition:Topologically Complemented Subspace of Banach Space",
"Linear Isomorphic Image of Complemented Subspace of Banach Space is Complemented",
"Definition:... |
proofwiki-22855 | Pitt's Theorem | Let $\GF \in \set {\R, \C}$.
Let $p, r \in \hointr 1 \infty$ be such that $p < r$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ and $\struct {\map {\ell_r} \GF, \norm {\, \cdot \,}_r}$ be the $p$ and $r$-sequence spaces.
Let $X$ be a closed vector subspace of $\struct {\map {\ell_r} \GF, \norm {\, \cdot \,... | Let $w$ denote the weak topology on $\struct {\map {\ell_r} \GF, \norm {\, \cdot \,}_r}$.
From P-Sequence Space is Reflexive for Finite p Greater Than One:
:$\struct {\map {\ell_r} \GF, \norm {\, \cdot \,}_r}$ is reflexive.
From Closed Vector Subspace of Reflexive Banach Space is Reflexive:
:$X$ is reflexive.
Let $B_X... | Let $\GF \in \set {\R, \C}$.
Let $p, r \in \hointr 1 \infty$ be such that $p < r$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ and $\struct {\map {\ell_r} \GF, \norm {\, \cdot \,}_r}$ be the [[Definition:P-Sequence Space|$p$ and $r$-sequence spaces]].
Let $X$ be a [[Definition:Closed Set|closed]] [[Def... | Let $w$ denote the [[Definition:Weak Topology|weak topology]] on $\struct {\map {\ell_r} \GF, \norm {\, \cdot \,}_r}$.
From [[P-Sequence Space is Reflexive for Finite p Greater Than One]]:
:$\struct {\map {\ell_r} \GF, \norm {\, \cdot \,}_r}$ is [[Definition:Reflexive Space|reflexive]].
From [[Closed Vector Subspace... | Pitt's Theorem | https://proofwiki.org/wiki/Pitt's_Theorem | https://proofwiki.org/wiki/Pitt's_Theorem | [
"Compact Linear Transformations",
"P-Sequence Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:Closed Set",
"Definition:Vector Subspace",
"Definition:Space of Bounded Linear Transformations",
"Definition:Space of Compact Linear Transformations",
"Definition:Bounded Linear Transformation",
"Definition:Compact Linear Transformation"
] | [
"Definition:Initial Topology",
"P-Sequence Space is Reflexive for Finite p Greater Than One",
"Definition:Reflexive Space",
"Closed Vector Subspace of Reflexive Banach Space is Reflexive",
"Definition:Reflexive Space",
"Definition:Closed Unit Ball",
"Kakutani's Theorem",
"Definition:Compact Topologica... |
proofwiki-22856 | Closed Vector Subspace of Reflexive Banach Space is Reflexive | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a reflexive Banach space over $\GF$.
Let $Y$ be a closed vector subspace of $X$.
Then $Y$ is reflexive. | Let $B_X^-$ be the closed unit ball in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_Y^-$ be the closed unit ball in $\struct {Y, \norm {\, \cdot \,}_X}$.
From Kakutani's Theorem, $B_X^-$ is weakly compact.
From Closed Unit Ball in Normed Vector Space is Weakly Closed, $B_Y^-$ is weakly closed in $Y$.
From Closed Vecto... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Reflexive Space|reflexive]] [[Definition:Banach Space|Banach space]] over $\GF$.
Let $Y$ be a [[Definition:Closed Set|closed]] [[Definition:Vector Subspace|vector subspace]] of $X$.
Then $Y$ is [[Definition:Reflexive Space|refle... | Let $B_X^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_Y^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $\struct {Y, \norm {\, \cdot \,}_X}$.
From [[Kakutani's Theorem]], $B_X^-$ is [[Definition:Weakly Compact Set|weakly compact]].
From [[C... | Closed Vector Subspace of Reflexive Banach Space is Reflexive | https://proofwiki.org/wiki/Closed_Vector_Subspace_of_Reflexive_Banach_Space_is_Reflexive | https://proofwiki.org/wiki/Closed_Vector_Subspace_of_Reflexive_Banach_Space_is_Reflexive | [
"Reflexive Spaces"
] | [
"Definition:Reflexive Space",
"Definition:Banach Space",
"Definition:Closed Set",
"Definition:Vector Subspace",
"Definition:Reflexive Space"
] | [
"Definition:Closed Unit Ball",
"Definition:Closed Unit Ball",
"Kakutani's Theorem",
"Definition:Weakly Compact Set",
"Closed Unit Ball in Normed Vector Space is Weakly Closed",
"Definition:Weakly Closed Set",
"Closed Vector Subspace in Normed Vector Space is Weakly Closed",
"Definition:Weakly Closed S... |
proofwiki-22857 | P-Sequence Space is Reflexive for Finite p Greater Than One | Let $\GF \in \set {\R, \C}$.
Let $p \in \openint 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the $p$-sequence space.
Then $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ is reflexive. | Let $\struct {\map {\ell_p^\ast} \GF, \norm {\, \cdot \,}_{\ell_p^\ast} }$ be the normed dual space of $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$.
Let $\struct {\map {\ell_q^\ast} \GF, \norm {\, \cdot \,}_{\ell_q^\ast} }$ be the normed dual space of $\struct {\map {\ell_q} \GF, \norm {\, \cdot \,}_q}$.
Let $\... | Let $\GF \in \set {\R, \C}$.
Let $p \in \openint 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the [[Definition:P-Sequence Space|$p$-sequence space]].
Then $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ is [[Definition:Reflexive Space|reflexive]]. | Let $\struct {\map {\ell_p^\ast} \GF, \norm {\, \cdot \,}_{\ell_p^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$.
Let $\struct {\map {\ell_q^\ast} \GF, \norm {\, \cdot \,}_{\ell_q^\ast} }$ be the [[Definition:Normed Dual Space|normed dual spac... | P-Sequence Space is Reflexive for Finite p Greater Than One | https://proofwiki.org/wiki/P-Sequence_Space_is_Reflexive_for_Finite_p_Greater_Than_One | https://proofwiki.org/wiki/P-Sequence_Space_is_Reflexive_for_Finite_p_Greater_Than_One | [
"Reflexive Spaces",
"P-Sequence Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:Reflexive Space"
] | [
"Definition:Normed Dual Space",
"Definition:Normed Dual Space",
"Definition:Second Normed Dual",
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation",
"Normed Dual Space of p-Sequence Space is Isometrically Isomorphic to q-Sequence Space",
"Normed Dual Space of p-Sequence Space... |
proofwiki-22858 | Rational Numbers form Subfield of Constructible Numbers | The rational numbers form a subfield of the constructible numbers. | From Constructible Numbers form Field, the constructible numbers form a field.
From Rational Numbers form Field, the rational numbers form a field.
Let $a \in \Q$ be a rational number.
From Construction of Point in Cartesian Plane with Rational Coordinates, $a$ is a constructible number.
Hence the rational numbers form... | The [[Definition:Rational Number|rational numbers]] form a [[Definition:Subfield|subfield]] of the [[Definition:Constructible Number|constructible numbers]]. | From [[Constructible Numbers form Field]], the [[Definition:Constructible Number|constructible numbers]] form a [[Definition:Field (Abstract Algebra)|field]].
From [[Rational Numbers form Field]], the [[Definition:Rational Number|rational numbers]] form a [[Definition:Field (Abstract Algebra)|field]].
Let $a \in \Q$ ... | Rational Numbers form Subfield of Constructible Numbers | https://proofwiki.org/wiki/Rational_Numbers_form_Subfield_of_Constructible_Numbers | https://proofwiki.org/wiki/Rational_Numbers_form_Subfield_of_Constructible_Numbers | [
"Constructible Numbers",
"Rational Numbers"
] | [
"Definition:Rational Number",
"Definition:Subfield",
"Definition:Constructible Number"
] | [
"Constructible Numbers form Field",
"Definition:Constructible Number",
"Definition:Field (Abstract Algebra)",
"Rational Numbers form Field",
"Definition:Rational Number",
"Definition:Field (Abstract Algebra)",
"Definition:Rational Number",
"Construction of Point in Cartesian Plane with Rational Coordi... |
proofwiki-22859 | Constructible Numbers form Subfield of Algebraic Numbers | The constructible numbers form a subfield of the algebraic numbers. | From Constructible Numbers form Field, the constructible numbers form a field.
From Algebraic Numbers form Field, the algebraic numbers form a field.
It remains to be shown that the constructible numbers form a subset of the algebraic numbers.
{{finish}} | The [[Definition:Constructible Number|constructible numbers]] form a [[Definition:Subfield|subfield]] of the [[Definition:Algebraic Number|algebraic numbers]]. | From [[Constructible Numbers form Field]], the [[Definition:Constructible Number|constructible numbers]] form a [[Definition:Field (Abstract Algebra)|field]].
From [[Algebraic Numbers form Field]], the [[Definition:Algebraic Number|algebraic numbers]] form a [[Definition:Field (Abstract Algebra)|field]].
It remains t... | Constructible Numbers form Subfield of Algebraic Numbers | https://proofwiki.org/wiki/Constructible_Numbers_form_Subfield_of_Algebraic_Numbers | https://proofwiki.org/wiki/Constructible_Numbers_form_Subfield_of_Algebraic_Numbers | [
"Constructible Numbers",
"Algebraic Numbers"
] | [
"Definition:Constructible Number",
"Definition:Subfield",
"Definition:Algebraic Number"
] | [
"Constructible Numbers form Field",
"Definition:Constructible Number",
"Definition:Field (Abstract Algebra)",
"Algebraic Numbers form Field",
"Definition:Algebraic Number",
"Definition:Field (Abstract Algebra)",
"Definition:Constructible Number",
"Definition:Subset",
"Definition:Algebraic Number"
] |
proofwiki-22860 | Weakly Compact Set in P-Sequence Space is Weakly Metrizable | Let $\GF \in \set {\R, \C}$.
Let $p \in \closedint 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the $p$-sequence space.
Let $w$ be the weak topology on $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$.
Let $K \subseteq \map {\ell_p} \GF$ be weakly compact.
Then $\struct {K, w}$ is metriza... | From P-Sequence Space with P-Norm forms Banach Space, $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ is a Banach space.
Let $\struct {\map {\ell_p^\ast} \GF, \norm {\, \cdot \,}_{\ell_p^\ast} }$ be the normed dual space of $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$.
Define $\pi_j : \map {\ell_p} \GF \to... | Let $\GF \in \set {\R, \C}$.
Let $p \in \closedint 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the [[Definition:P-Sequence Space|$p$-sequence space]].
Let $w$ be the [[Definition:Weak Topology|weak topology]] on $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$.
Let $K \subseteq \map ... | From [[P-Sequence Space with P-Norm forms Banach Space]], $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ is a [[Definition:Banach Space|Banach space]].
Let $\struct {\map {\ell_p^\ast} \GF, \norm {\, \cdot \,}_{\ell_p^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {\map {\ell_p} \... | Weakly Compact Set in P-Sequence Space is Weakly Metrizable | https://proofwiki.org/wiki/Weakly_Compact_Set_in_P-Sequence_Space_is_Weakly_Metrizable | https://proofwiki.org/wiki/Weakly_Compact_Set_in_P-Sequence_Space_is_Weakly_Metrizable | [
"Metrizable Spaces",
"Weakly Compact Sets",
"P-Sequence Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:Initial Topology",
"Definition:Weakly Compact Set",
"Definition:Metrizable Space"
] | [
"P-Sequence Space with P-Norm forms Banach Space",
"Definition:Banach Space",
"Definition:Normed Dual Space",
"Definition:Mappings Separating Points",
"Definition:Mappings Separating Points",
"Definition:Initial Topology",
"Definition:Mappings Separating Points",
"Definition:Countable Set",
"Definit... |
proofwiki-22861 | Not every Algebraic Number is Constructible | There exist algebraic numbers which are not constructible. | Let $a$ be a constructible number.
From Constructible Length with Compass and Straightedge, $a$ is algebraic with degree equal to a power of $2$.
So, for example, $\sqrt [3] 2$ is not constructible, although it is an algebraic number, as a root of the equation $x^3 = 2$.
{{qed}} | There exist [[Definition:Algebraic Number|algebraic numbers]] which are not [[Definition:Constructible Number|constructible]]. | Let $a$ be a [[Definition:Constructible Number|constructible number]].
From [[Constructible Length with Compass and Straightedge]], $a$ is [[Definition:Algebraic Number|algebraic]] with [[Definition:Degree of Algebraic Number|degree]] equal to a [[Definition:Integer Power|power]] of $2$.
So, for example, $\sqrt [3] ... | Not every Algebraic Number is Constructible | https://proofwiki.org/wiki/Not_every_Algebraic_Number_is_Constructible | https://proofwiki.org/wiki/Not_every_Algebraic_Number_is_Constructible | [
"Constructible Numbers",
"Algebraic Numbers"
] | [
"Definition:Algebraic Number",
"Definition:Constructible Number"
] | [
"Definition:Constructible Number",
"Constructible Length with Compass and Straightedge",
"Definition:Algebraic Number",
"Definition:Algebraic Number/Degree",
"Definition:Power (Algebra)/Integer",
"Definition:Constructible Number",
"Definition:Algebraic Number",
"Definition:Root of Polynomial",
"Defi... |
proofwiki-22862 | Equivalence of Basic Sequences in Banach Spaces is Transitive | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$, $\struct {Y, \norm {\, \cdot \,}_Y}$ and $\struct {Z, \norm {\, \cdot \,}_Z}$ be Banach spaces over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a basic sequence in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\sequence {y_n}_{n \mathop \in \N}$... | Since $\sequence {x_n}_{n \mathop \in \N}$ is equivalent to $\sequence {y_n}_{n \mathop \in \N}$, we have that:
:for every sequence $\sequence {\alpha_n}_{n \mathop \in \N}$ in $\GF$ we have:
::$\ds \sum_{j \mathop = 1}^\infty \alpha_j x_j$ converges in $\struct {X, \norm {\, \cdot \,}_X}$ {{iff}} $\ds \sum_{j \mathop ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$, $\struct {Y, \norm {\, \cdot \,}_Y}$ and $\struct {Z, \norm {\, \cdot \,}_Z}$ 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... | Since $\sequence {x_n}_{n \mathop \in \N}$ is [[Definition:Equivalence of Basic Sequences|equivalent]] to $\sequence {y_n}_{n \mathop \in \N}$, we have that:
:for every [[Definition:Sequence|sequence]] $\sequence {\alpha_n}_{n \mathop \in \N}$ in $\GF$ we have:
::$\ds \sum_{j \mathop = 1}^\infty \alpha_j x_j$ [[Definit... | Equivalence of Basic Sequences in Banach Spaces is Transitive | https://proofwiki.org/wiki/Equivalence_of_Basic_Sequences_in_Banach_Spaces_is_Transitive | https://proofwiki.org/wiki/Equivalence_of_Basic_Sequences_in_Banach_Spaces_is_Transitive | [
"Basic Sequences"
] | [
"Definition:Banach Space",
"Definition:Basic Sequence",
"Definition:Basic Sequence",
"Definition:Equivalence of Basic Sequences",
"Definition:Basic Sequence",
"Definition:Equivalence of Basic Sequences",
"Definition:Equivalence of Basic Sequences"
] | [
"Definition:Equivalence of Basic Sequences",
"Definition:Sequence",
"Definition:Convergent Sequence",
"Definition:Convergent Sequence",
"Definition:Equivalence of Basic Sequences",
"Definition:Sequence",
"Definition:Convergent Sequence",
"Definition:Convergent Sequence",
"Definition:Sequence",
"De... |
proofwiki-22863 | Composition of Linear Isomorphisms is Linear Isomorphism | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$, $\struct {Y, \norm {\, \cdot \,}_Y}$ and $\struct {Z, \norm {\, \cdot \,}_Z}$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ and $S : Y \to Z$ be linear isomorphisms.
Then $S T : X \to Z$ is a linear isomorphism. | From Composition of Linear Transformations is Linear Transformation, $S T$ is a linear transformation.
We have that $T$, $T^{-1}$, $S$ and $S^{-1}$ are bounded.
From Composition of Bounded Linear Transformations is Bounded Linear Transformation, $S T$ and $T^{-1} S^{-1}$ are bounded.
From Inverse of Composite Bijectio... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$, $\struct {Y, \norm {\, \cdot \,}_Y}$ and $\struct {Z, \norm {\, \cdot \,}_Z}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ and $S : Y \to Z$ be [[Definition:Linear Isomorphism|linear isomorphisms]].
Th... | From [[Composition of Linear Transformations is Linear Transformation]], $S T$ is a [[Definition:Linear Transformation|linear transformation]].
We have that $T$, $T^{-1}$, $S$ and $S^{-1}$ are [[Definition:Bounded Linear Transformation|bounded]].
From [[Composition of Bounded Linear Transformations is Bounded Linear... | Composition of Linear Isomorphisms is Linear Isomorphism | https://proofwiki.org/wiki/Composition_of_Linear_Isomorphisms_is_Linear_Isomorphism | https://proofwiki.org/wiki/Composition_of_Linear_Isomorphisms_is_Linear_Isomorphism | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Linear Isomorphism",
"Definition:Linear Isomorphism"
] | [
"Composition of Linear Transformations is Linear Transformation",
"Definition:Linear Transformation",
"Definition:Bounded Linear Transformation",
"Norm on Bounded Linear Transformation is Submultiplicative",
"Definition:Bounded Linear Transformation",
"Inverse of Composite Bijection",
"Definition:Bounde... |
proofwiki-22864 | Complement of Prime Ideal is Prime Filter | Let $P = \struct {S, \preceq}$ be an ordered set.
Let $I \subseteq S$.
Then:
:$I$ is a prime ideal of $P$
{{iff}}:
:$S \setminus I$ is a prime filter of $P$. | Since $S \setminus \paren {S \setminus I} = I$, both conditions are equivalent by definition to the conjunction of:
:$I$ is an ideal of $P$
:$S \setminus I$ is a filter of $P$
The result follows.
{{qed}}
Category:Prime Ideals (Order Theory)
m5mh9pbryrga986lpw9ufxg3rtbmzz8 | Let $P = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $I \subseteq S$.
Then:
:$I$ is a [[Definition:Prime Ideal (Order Theory)|prime ideal]] of $P$
{{iff}}:
:$S \setminus I$ is a [[Definition:Prime Filter (Order Theory)|prime filter]] of $P$. | Since $S \setminus \paren {S \setminus I} = I$, both conditions are equivalent by definition to the [[Definition:Conjunction|conjunction]] of:
:$I$ is an [[Definition:Ideal (Order Theory)|ideal]] of $P$
:$S \setminus I$ is a [[Definition:Filter|filter]] of $P$
The result follows.
{{qed}}
[[Category:Prime Ideals (Orde... | Complement of Prime Ideal is Prime Filter | https://proofwiki.org/wiki/Complement_of_Prime_Ideal_is_Prime_Filter | https://proofwiki.org/wiki/Complement_of_Prime_Ideal_is_Prime_Filter | [
"Prime Ideals (Order Theory)"
] | [
"Definition:Ordered Set",
"Definition:Prime Ideal (Order Theory)",
"Definition:Prime Filter (Order Theory)"
] | [
"Definition:Conjunction",
"Definition:Ideal (Order Theory)",
"Definition:Filter",
"Category:Prime Ideals (Order Theory)"
] |
proofwiki-22865 | Complement of Prime Filter is Prime Ideal | Let $P = \struct {S, \preceq}$ be an ordered set.
Let $F \subseteq S$.
Then:
:$F$ is a prime filter of $P$
{{iff}}:
:$S \setminus F$ is a prime ideal of $P$. | Let $P^{-1}$ denote the dual of $P$.
By Prime Filter is Prime Ideal in Dual Ordered Set:
:$F$ is a prime filter of $P$
{{iff}}:
:$F$ is a prime ideal of $P^{-1}$
By Prime Ideal is Prime Filter in Dual Ordered Set:
:$S \setminus F$ is a prime ideal of $P$
{{iff}}:
:$S \setminus F$ is a prime filter of $P^{-1}$
The resul... | Let $P = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $F \subseteq S$.
Then:
:$F$ is a [[Definition:Prime Filter (Order Theory)|prime filter]] of $P$
{{iff}}:
:$S \setminus F$ is a [[Definition:Prime Ideal (Order Theory)|prime ideal]] of $P$. | Let $P^{-1}$ denote the [[Definition:Dual Ordered Set|dual]] of $P$.
By [[Prime Filter is Prime Ideal in Dual Ordered Set]]:
:$F$ is a [[Definition:Prime Filter (Order Theory)|prime filter]] of $P$
{{iff}}:
:$F$ is a [[Definition:Prime Ideal (Order Theory)|prime ideal]] of $P^{-1}$
By [[Prime Ideal is Prime Filter ... | Complement of Prime Filter is Prime Ideal | https://proofwiki.org/wiki/Complement_of_Prime_Filter_is_Prime_Ideal | https://proofwiki.org/wiki/Complement_of_Prime_Filter_is_Prime_Ideal | [
"Prime Ideals (Order Theory)"
] | [
"Definition:Ordered Set",
"Definition:Prime Filter (Order Theory)",
"Definition:Prime Ideal (Order Theory)"
] | [
"Definition:Dual Ordering/Dual Ordered Set",
"Prime Filter is Prime Ideal in Dual Ordered Set",
"Definition:Prime Filter (Order Theory)",
"Definition:Prime Ideal (Order Theory)",
"Prime Ideal is Prime Filter in Dual Ordered Set",
"Definition:Prime Ideal (Order Theory)",
"Definition:Prime Filter (Order T... |
proofwiki-22866 | Equivalence of Basic Sequences in Normed Vector Spaces is Preserved by Scalar Multiplication | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a basic sequence in $X$.
Let $\sequence {y_n}_{n \mathop \in \N}$ be a basic sequence in $Y$ that is equivalent to $\sequence {x_... | From Scalar Multiple of Basic Sequence is Basic, both $\sequence {\lambda_n x_n}_{n \mathop \in \N}$ and $\sequence {\lambda_n y_n}_{n \mathop \in \N}$ are basic.
Let $\sequence {\alpha_n}_{n \mathop \in \N}$ be a sequence in $\GF$.
Since $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ are... | 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]] over $\GF$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Basic Sequence|basic sequence]] in $X$.
Let $\sequence {y_n}_{n \mathop \in ... | From [[Scalar Multiple of Basic Sequence is Basic]], both $\sequence {\lambda_n x_n}_{n \mathop \in \N}$ and $\sequence {\lambda_n y_n}_{n \mathop \in \N}$ are [[Definition:Basic Sequence|basic]].
Let $\sequence {\alpha_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\GF$.
Since $\sequence {x_n}_{n \... | Equivalence of Basic Sequences in Normed Vector Spaces is Preserved by Scalar Multiplication | https://proofwiki.org/wiki/Equivalence_of_Basic_Sequences_in_Normed_Vector_Spaces_is_Preserved_by_Scalar_Multiplication | https://proofwiki.org/wiki/Equivalence_of_Basic_Sequences_in_Normed_Vector_Spaces_is_Preserved_by_Scalar_Multiplication | [
"Basic Sequences"
] | [
"Definition:Normed Vector Space",
"Definition:Basic Sequence",
"Definition:Basic Sequence",
"Definition:Equivalence of Basic Sequences",
"Definition:Sequence",
"Definition:Equivalence of Basic Sequences"
] | [
"Scalar Multiple of Basic Sequence is Basic",
"Definition:Basic Sequence",
"Definition:Sequence",
"Definition:Equivalence of Basic Sequences",
"Definition:Sequence",
"Definition:Convergent Series",
"Definition:Convergent Series",
"Definition:Convergent Series",
"Definition:Convergent Series",
"Def... |
proofwiki-22867 | Characterization of Bounded Linear Transformation from Normed Vector Space to Space of Bounded Sequences | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$ such that $X \ne \set { {\mathbf 0}_X}$.
Let $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$ be the space of bounded sequences.
Let $T : X \to \map {\ell_\infty} \GF$ be a mapping.
Let $\struct {X^\a... | Let $\struct {\map {\ell_\infty^\ast} \GF, \norm {\, \cdot \,}_{\ell_\infty^\ast} }$ be the normed dual space of $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$.
For each $j \in \N$, define $\pi_j : \map {\ell_\infty} \GF \to \GF$ by:
:$\map {\pi_j} x = x_j$ for each $x \in \map {\ell_\infty} \GF$.
From ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$ such that $X \ne \set { {\mathbf 0}_X}$.
Let $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Bounded Sequences|space of bounded seque... | Let $\struct {\map {\ell_\infty^\ast} \GF, \norm {\, \cdot \,}_{\ell_\infty^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$.
For each $j \in \N$, define $\pi_j : \map {\ell_\infty} \GF \to \GF$ by:
:$\map {\pi_j} x = x_j$ for each $x ... | Characterization of Bounded Linear Transformation from Normed Vector Space to Space of Bounded Sequences | https://proofwiki.org/wiki/Characterization_of_Bounded_Linear_Transformation_from_Normed_Vector_Space_to_Space_of_Bounded_Sequences | https://proofwiki.org/wiki/Characterization_of_Bounded_Linear_Transformation_from_Normed_Vector_Space_to_Space_of_Bounded_Sequences | [
"Space of Bounded Sequences"
] | [
"Definition:Normed Vector Space",
"Definition:Space of Bounded Sequences",
"Definition:Mapping",
"Definition:Normed Dual Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Bounded Linear Transformation",
"Definition:Sequence",
"Definition:Bounded Linear Functional",
"Definitio... | [
"Definition:Normed Dual Space",
"Coordinate Projection on P-Sequence Space is Bounded Linear Functional"
] |
proofwiki-22868 | Contact Forces are Equal and Opposite | Let $A$ and $B$ be bodies in contact.
The contact force of $A$ on $B$ is equal in magnitude and in the opposite direction to the contact force of $B$ on $A$. | This is a direct consequence of Newton's Third Law of Motion. | Let $A$ and $B$ be [[Definition:Body|bodies]] in [[Definition:Contact|contact]].
The [[Definition:Contact Force|contact force]] of $A$ on $B$ is equal in [[Definition:Magnitude|magnitude]] and in the [[Definition:Opposite Direction|opposite direction]] to the [[Definition:Contact Force|contact force]] of $B$ on $A$. | This is a direct consequence of [[Newton's Third Law of Motion]]. | Contact Forces are Equal and Opposite | https://proofwiki.org/wiki/Contact_Forces_are_Equal_and_Opposite | https://proofwiki.org/wiki/Contact_Forces_are_Equal_and_Opposite | [
"Contact Forces"
] | [
"Definition:Body",
"Definition:Contact",
"Definition:Contact Force",
"Definition:Magnitude",
"Definition:Opposite Direction",
"Definition:Contact Force"
] | [
"Newton's Laws of Motion/Third Law"
] |
proofwiki-22869 | Infinite-Dimensional Vector Subspace of P-Sequence Space contains Topologically Complemented Subspace Linearly Isomorphic to Whole Space | Let $\GF \in \set {\R, \C}$.
Let $p \in \hointr 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the $p$-sequence space.
Let $Y$ be an infinite-dimensional closed vector subspace of $\map {\ell_p} \GF$.
Then there exists a closed vector subspace $Z$ of $\map {\ell_p} \GF$ that is topologically com... | 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.
Let $\sequence {e_n^\ast}_{n \mathop \in \N}$ be the coordinate functionals associated with $\sequence {e_n}_{n \mathop \in \N}$.
We show that, for each $n \in \N$, we can find $x_n \i... | Let $\GF \in \set {\R, \C}$.
Let $p \in \hointr 1 \infty$.
Let $\struct {\map {\ell_p} \GF, \norm {\, \cdot \,}_p}$ be the [[Definition:P-Sequence Space|$p$-sequence space]].
Let $Y$ be an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Closed Set|closed]] [[Definition:Vector Subsp... | 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]].
Let $\sequence {e_n^\ast}_{n \mathop \in \N}$ be the [[Definition:Coordinate Functionals Associated with Schauder Basis|coordinate functionals]] asso... | Infinite-Dimensional Vector Subspace of P-Sequence Space contains Topologically Complemented Subspace Linearly Isomorphic to Whole Space | https://proofwiki.org/wiki/Infinite-Dimensional_Vector_Subspace_of_P-Sequence_Space_contains_Topologically_Complemented_Subspace_Linearly_Isomorphic_to_Whole_Space | https://proofwiki.org/wiki/Infinite-Dimensional_Vector_Subspace_of_P-Sequence_Space_contains_Topologically_Complemented_Subspace_Linearly_Isomorphic_to_Whole_Space | [
"P-Sequence Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:Infinite-Dimensional Vector Space",
"Definition:Closed Set",
"Definition:Vector Subspace",
"Definition:Closed Set",
"Definition:Vector Subspace",
"Definition:Topologically Complemented Subspace of Banach Space",
"Definition:Linear Isomorphism"
] | [
"Definition:Schauder Basis",
"P-Sequence Space admits Schauder Basis",
"Definition:Coordinate Functionals Associated with Schauder Basis",
"Definition:Projections Associated with Schauder Basis",
"Coordinate Functionals Associated with Schauder Basis of Banach Space are Bounded/Corollary 1",
"Definition:B... |
proofwiki-22870 | Composite of Continuous Real Functions at Point is Continuous | Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions.
Let $g$ be continuous at a point $a \in \R$.
Let $f$ be continuous at the point $\map g a \in \R$.
Then their composite $f \circ g$ is continuous at $a$. | This follows from Limit of Composite Function.
Indeed, we have, from continuity hypotheses:
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \rightarrow a} \map g x
| r = \map g a
| c =
}}
{{eqn | l = \lim_{y \mathop \rightarrow \map g a} \map f y
| r = \map f {\map g a}
| c =
}}
{{eqn | ll= \leadsto... | Let $\R$ denote the [[Definition:Real Numbers|real numbers]].
Let $f$ and $g$ be [[Definition:Real Function|real functions]].
Let $g$ be [[Definition:Continuous Real Function at Point|continuous]] at a [[Definition:Point|point]] $a \in \R$.
Let $f$ be [[Definition:Continuous Real Function at Point|continuous]] at t... | This follows from [[Limit of Composite Function]].
Indeed, we have, from continuity hypotheses:
{{begin-eqn}}
{{eqn | l = \lim_{x \mathop \rightarrow a} \map g x
| r = \map g a
| c =
}}
{{eqn | l = \lim_{y \mathop \rightarrow \map g a} \map f y
| r = \map f {\map g a}
| c =
}}
{{eqn | ll= \l... | Composite of Continuous Real Functions at Point is Continuous | https://proofwiki.org/wiki/Composite_of_Continuous_Real_Functions_at_Point_is_Continuous | https://proofwiki.org/wiki/Composite_of_Continuous_Real_Functions_at_Point_is_Continuous | [
"Continuous Real Functions"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Continuous Real Function/Point",
"Definition:Point",
"Definition:Continuous Real Function/Point",
"Definition:Point",
"Definition:Composition of Mappings",
"Definition:Continuous Real Function"
] | [
"Limit of Composite Function"
] |
proofwiki-22871 | Space of Bounded Sequences Supported on Subset of Natural Numbers is Closed Vector Subspace of Space of Bounded Sequences | Let $\GF \in \set {\R, \C}$.
Let $A \subseteq \N$.
Let $\struct {\map {\ell_\infty} {\N, \GF}, \norm {\, \cdot \,}_\infty}$ be the space of bounded sequences.
Let:
:$\map {\ell_\infty} {A, \GF} = \set {\xi \in \map {\ell_\infty} \N : \map \xi n = 0 \text { for } n \in \N \setminus A}$
Equip $\map {\ell_\infty} {A, \GF... | === $\map {\ell_\infty} A$ is closed ===
Let $\sequence {\xi_n}_{n \mathop \in \N}$ be a sequence of elements of $\map {\ell_\infty} {A, \GF}$ converging to $\xi$.
Then:
:$\map {\xi_n} j = 0$ for $j \in \N \setminus A$
We have:
:$\cmod {\map \xi j} = \cmod {\map {\xi_n} j - \map \xi j} \le \norm {\xi_n - \xi}_\infty$ ... | Let $\GF \in \set {\R, \C}$.
Let $A \subseteq \N$.
Let $\struct {\map {\ell_\infty} {\N, \GF}, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Bounded Sequences|space of bounded sequences]].
Let:
:$\map {\ell_\infty} {A, \GF} = \set {\xi \in \map {\ell_\infty} \N : \map \xi n = 0 \text { for } n \in \N \s... | === $\map {\ell_\infty} A$ is [[Definition:Closed Set|closed]] ===
Let $\sequence {\xi_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of elements of $\map {\ell_\infty} {A, \GF}$ [[Definition:Convergent Sequence|converging]] to $\xi$.
Then:
:$\map {\xi_n} j = 0$ for $j \in \N \setminus A$
We have:
:$\... | Space of Bounded Sequences Supported on Subset of Natural Numbers is Closed Vector Subspace of Space of Bounded Sequences | https://proofwiki.org/wiki/Space_of_Bounded_Sequences_Supported_on_Subset_of_Natural_Numbers_is_Closed_Vector_Subspace_of_Space_of_Bounded_Sequences | https://proofwiki.org/wiki/Space_of_Bounded_Sequences_Supported_on_Subset_of_Natural_Numbers_is_Closed_Vector_Subspace_of_Space_of_Bounded_Sequences | [
"Space of Bounded Sequences"
] | [
"Definition:Space of Bounded Sequences",
"Definition:Ring of Sequences/Pointwise Addition",
" Definition:Pointwise Multiplication on Ring of Sequences",
"Definition:Closed Set",
"Definition:Vector Subspace"
] | [
"Definition:Closed Set",
"Definition:Sequence",
"Definition:Convergent Sequence",
"Definition:Convergent Sequence",
"Definition:Limit of Sequence",
"Definition:Closed Set"
] |
proofwiki-22872 | Bounded Linear Operator on Space of Bounded Sequences Vanishing on Zero-Limit Sequences also Vanishes on Bounded Sequences with certain Support | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {\ell_\infty} {\N, \GF}, \norm {\, \cdot \,}_\infty}$ be the space of bounded sequences.
Let $\struct {\map {c_0} {\N, \GF}, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
Let $T : \map {\ell_\infty} {\N, \GF} \to \map {\ell_\infty} {\N, \GF}$ be a bo... | Let $\struct {\map \BB {\map {\ell_\infty} {\N, \GF} }, \norm {\, \cdot \,}_{\map \BB {\ell_\infty} } }$ be the space of bounded linear transformations $\map {\ell_\infty} {\N, \GF} \to \map {\ell_\infty} {\N, \GF}$.
From Countably Infinite Set has Uncountable Family of Subsets with Finite Intersection, there exists a... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {\ell_\infty} {\N, \GF}, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Bounded Sequences|space of bounded sequences]].
Let $\struct {\map {c_0} {\N, \GF}, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Zero-Limit Sequences|space of zero-limit sequ... | Let $\struct {\map \BB {\map {\ell_\infty} {\N, \GF} }, \norm {\, \cdot \,}_{\map \BB {\ell_\infty} } }$ be the [[Definition:Space of Bounded Linear Transformations|space of bounded linear transformations]] $\map {\ell_\infty} {\N, \GF} \to \map {\ell_\infty} {\N, \GF}$.
From [[Countably Infinite Set has Uncountable ... | Bounded Linear Operator on Space of Bounded Sequences Vanishing on Zero-Limit Sequences also Vanishes on Bounded Sequences with certain Support | https://proofwiki.org/wiki/Bounded_Linear_Operator_on_Space_of_Bounded_Sequences_Vanishing_on_Zero-Limit_Sequences_also_Vanishes_on_Bounded_Sequences_with_certain_Support | https://proofwiki.org/wiki/Bounded_Linear_Operator_on_Space_of_Bounded_Sequences_Vanishing_on_Zero-Limit_Sequences_also_Vanishes_on_Bounded_Sequences_with_certain_Support | [
"Space of Bounded Sequences",
"Space of Zero-Limit Sequences"
] | [
"Definition:Space of Bounded Sequences",
"Definition:Space of Zero-Limit Sequences",
"Definition:Bounded Linear Transformation",
"Definition:Countably Infinite/Set",
"Definition:Set",
"Definition:Space of Bounded Sequences Supported on Subset of Natural Numbers"
] | [
"Definition:Space of Bounded Linear Transformations",
"Countably Infinite Set has Uncountable Family of Subsets with Finite Intersection",
"Definition:Uncountable/Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Finite Set",
"Definition:Countable Set",
"Countable Union of Countable Sets is Co... |
proofwiki-22873 | Phillips-Sobczyk Theorem | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {\ell_\infty} {\N, \GF}, \norm {\, \cdot \,}_\infty}$ be the space of bounded sequences.
Let $\struct {\map {c_0} {\N, \GF}, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
Then there does not exist a bounded projection $P : \map {\ell_\infty} {\N, \GF}... | {{AimForCont}} there exists a bounded projection $P : \map {\ell_\infty} {\N, \GF} \to \map {c_0} {\N, \GF}$.
Let $I : \map {\ell_\infty} {\N, \GF} \to \map {\ell_\infty} {\N, \GF}$ be the identity mapping.
Then $T = I - P$ is a bounded linear transformation.
Since $P$ is a projection onto $\map {c_0} {\N, \GF}$, we ha... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {\ell_\infty} {\N, \GF}, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Bounded Sequences|space of bounded sequences]].
Let $\struct {\map {c_0} {\N, \GF}, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Zero-Limit Sequences|space of zero-limit sequ... | {{AimForCont}} there exists a [[Definition:Bounded Linear Transformation|bounded]] [[Definition:Projection (Vector Spaces)|projection]] $P : \map {\ell_\infty} {\N, \GF} \to \map {c_0} {\N, \GF}$.
Let $I : \map {\ell_\infty} {\N, \GF} \to \map {\ell_\infty} {\N, \GF}$ be the [[Definition:Identity Mapping|identity mapp... | Phillips-Sobczyk Theorem | https://proofwiki.org/wiki/Phillips-Sobczyk_Theorem | https://proofwiki.org/wiki/Phillips-Sobczyk_Theorem | [
"Topologically Complemented Subspaces of Banach Spaces",
"Space of Bounded Sequences",
"Space of Zero-Limit Sequences"
] | [
"Definition:Space of Bounded Sequences",
"Definition:Space of Zero-Limit Sequences",
"Definition:Bounded Linear Transformation",
"Definition:Projection (Vector Spaces)",
"Definition:Topologically Complemented Subspace of Banach Space"
] | [
"Definition:Bounded Linear Transformation",
"Definition:Projection (Vector Spaces)",
"Definition:Identity Mapping",
"Definition:Bounded Linear Transformation",
"Definition:Projection (Vector Spaces)",
"Bounded Linear Operator on Space of Bounded Sequences Vanishing on Zero-Limit Sequences also Vanishes on... |
proofwiki-22874 | Weak-* Compact Set in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable | Let $X$ be a separable normed vector space.
Let $\sequence {x_n}_{n \in \N}$ be a countable everywhere dense subset of $X$.
Let $X^\ast$ be the normed dual space of $X$.
Let $w^\ast$ be the weak-$\ast$ topology on $X$.
Let $\struct {K, w^\ast}$ be a compact subspace of $X^\ast$.
Define $d : X^\ast \times X^\ast \to \R... | Let $\sequence {x_n^\wedge }_{n \in \N}$ be the evaluation maps associated with $\sequence {x_n}_{n \in \N}$.
From Image of Everywhere Dense Subset of Normed Vector Space Separates Points of Normed Dual Space, $\sequence {x_n^\wedge }_{n \in \N}$ separates the points of $X^\ast$.
Let $\sigma$ be the initial topology ge... | Let $X$ be a [[Definition:Separable Space|separable]] [[Definition:Normed Vector Space|normed vector space]].
Let $\sequence {x_n}_{n \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere dense subset]] of $X$.
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual spac... | Let $\sequence {x_n^\wedge }_{n \in \N}$ be the [[Definition:Evaluation Linear Transformation on Normed Vector Space|evaluation maps]] associated with $\sequence {x_n}_{n \in \N}$.
From [[Image of Everywhere Dense Subset of Normed Vector Space Separates Points of Normed Dual Space]], $\sequence {x_n^\wedge }_{n \in \N... | Weak-* Compact Set in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable | https://proofwiki.org/wiki/Weak-*_Compact_Set_in_Normed_Dual_Space_of_Separable_Normed_Vector_Space_is_Weak-*_Metrizable | https://proofwiki.org/wiki/Weak-*_Compact_Set_in_Normed_Dual_Space_of_Separable_Normed_Vector_Space_is_Weak-*_Metrizable | [
"Weak-* Topologies",
"Metrizable Spaces"
] | [
"Definition:Separable Space",
"Definition:Normed Vector Space",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Definition:Compact Topological Space/Subspace",
"Initial Topology Generated by Countable Family of Functions Separat... | [
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Image of Everywhere Dense Subset of Normed Vector Space Separates Points of Normed Dual Space",
"Definition:Mappings Separating Points",
"Definition:Initial Topology",
"Initial Topology Generated by Countable Family of Functions Separating ... |
proofwiki-22875 | Sobczyk's Theorem | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a separable Banach space over $\GF$.
Let $E$ be a closed vector subspace of $X$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
Let $\struct {\map \BB {E, c_0}, \norm {\, \cdot \,}_{\map \BB {E, c... | Note that if $T = {\mathbf 0}_{\map \BB {E, c_0} }$, then we can simply set:
:$\widetilde T x = {\mathbf 0}_{c_0}$ for each $x \in X$.
Then $\widetilde T$ extends $T$ and has:
:$0 = \norm {\widetilde T}_{\map \BB {X, c_0} } \le 2 \norm T_{\map \BB {E, c_0} } = 0$
We now initially assume that $\norm T_{\map \BB {E, c_... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Separable Space|separable]] [[Definition:Banach Space|Banach space]] over $\GF$.
Let $E$ be a [[Definition:Closed Set|closed]] [[Definition:Vector Subspace|vector subspace]] of $X$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot ... | Note that if $T = {\mathbf 0}_{\map \BB {E, c_0} }$, then we can simply set:
:$\widetilde T x = {\mathbf 0}_{c_0}$ for each $x \in X$.
Then $\widetilde T$ [[Definition:Extension of Mapping|extends]] $T$ and has:
:$0 = \norm {\widetilde T}_{\map \BB {X, c_0} } \le 2 \norm T_{\map \BB {E, c_0} } = 0$
We now initially... | Sobczyk's Theorem | https://proofwiki.org/wiki/Sobczyk's_Theorem | https://proofwiki.org/wiki/Sobczyk's_Theorem | [
"Separably Injective Banach Spaces",
"Space of Zero-Limit Sequences"
] | [
"Definition:Separable Space",
"Definition:Banach Space",
"Definition:Closed Set",
"Definition:Vector Subspace",
"Definition:Space of Zero-Limit Sequences",
"Definition:Space of Bounded Linear Transformations",
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation",
"D... | [
"Definition:Extension of Mapping",
"Definition:Normed Dual Space",
"Definition:Normed Dual Space",
"Characterization of Bounded Linear Transformation from Normed Vector Space to Space of Bounded Sequences",
"Definition:Sequence",
"Hahn-Banach Theorem",
"Definition:Extension of Mapping",
"Definition:We... |
proofwiki-22876 | Distance from Point to Compact Set in Metric Space is Attained | Let $\struct {X, d}$ be a metric space.
Let $A \subseteq X$ be compact.
Let $\map d {\cdot, A}$ denote the $d$-distance to $A$.
Let $x \in X$.
Then there exists $\alpha \in A$ such that:
:$\map d {x, A} = \map d {x, \alpha}$ | From Distance Function of Metric Space is Continuous, the mapping:
:$\tuple {x, a} \mapsto \map d {x, a}$ is continuous.
From Vertical Section of Continuous Function is Continuous, the mapping $f : A \to \hointr 0 \infty$ defined by:
:$\map f a = \map d {x, a}$ for each $a \in A$
is continuous.
From {{Corollary|Continu... | Let $\struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $A \subseteq X$ be [[Definition:Compact Topological Space|compact]].
Let $\map d {\cdot, A}$ denote the [[Definition:Distance between Element and Subset of Metric Space|$d$-distance to $A$]].
Let $x \in X$.
Then there exists $\alpha \in A$ suc... | From [[Distance Function of Metric Space is Continuous]], the [[Definition:Mapping|mapping]]:
:$\tuple {x, a} \mapsto \map d {x, a}$ is [[Definition:Continuous Function|continuous]].
From [[Vertical Section of Continuous Function is Continuous]], the [[Definition:Mapping|mapping]] $f : A \to \hointr 0 \infty$ defined ... | Distance from Point to Compact Set in Metric Space is Attained | https://proofwiki.org/wiki/Distance_from_Point_to_Compact_Set_in_Metric_Space_is_Attained | https://proofwiki.org/wiki/Distance_from_Point_to_Compact_Set_in_Metric_Space_is_Attained | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Compact Topological Space",
"Definition:Distance/Sets/Metric Spaces"
] | [
"Distance Function of Metric Space is Continuous",
"Definition:Mapping",
"Definition:Continuous Function",
"Vertical Section of Continuous Function is Continuous",
"Definition:Mapping",
"Definition:Continuous Function",
"Definition:Bounded Mapping",
"Category:Metric Spaces"
] |
proofwiki-22877 | Join of Meets Precedes Meet with Join | Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.
Let $x, y, z \in S$.
Then:
:$\paren {x \wedge y} \vee \paren {x \wedge z} \preceq x \wedge \paren {y \vee z}$ | By Meet Precedes Operands and Join Succeeds Operands:
:$x \wedge y \preceq x$
:$x \wedge z \preceq x$
:$x \wedge y \preceq y \preceq y \vee z$
:$x \wedge z \preceq z \preceq y \vee z$
so by definition of join:
:$\paren {x \wedge y} \vee \paren {x \wedge z} \preceq x$
:$\paren {x \wedge y} \vee \paren {x \wedge z} \prec... | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Lattice (Order Theory)|lattice]].
Let $x, y, z \in S$.
Then:
:$\paren {x \wedge y} \vee \paren {x \wedge z} \preceq x \wedge \paren {y \vee z}$ | By [[Meet Precedes Operands]] and [[Join Succeeds Operands]]:
:$x \wedge y \preceq x$
:$x \wedge z \preceq x$
:$x \wedge y \preceq y \preceq y \vee z$
:$x \wedge z \preceq z \preceq y \vee z$
so by definition of [[Definition:Join (Order Theory)|join]]:
:$\paren {x \wedge y} \vee \paren {x \wedge z} \preceq x$
:$\paren ... | Join of Meets Precedes Meet with Join | https://proofwiki.org/wiki/Join_of_Meets_Precedes_Meet_with_Join | https://proofwiki.org/wiki/Join_of_Meets_Precedes_Meet_with_Join | [
"Lattices (Order Theory)"
] | [
"Definition:Lattice (Order Theory)"
] | [
"Meet Precedes Operands",
"Join Succeeds Operands",
"Definition:Join (Order Theory)",
"Definition:Meet (Order Theory)",
"Category:Lattices (Order Theory)"
] |
proofwiki-22878 | Series whose Terms Converge to Zero is not necessarily Convergent | Let $\sequence {a_n}$ be a sequence in any of the standard number fields $\Q$, $\R$, or $\C$.
Suppose that $\sequence {a_n}$ converges to zero.
Then it is not necessarily the case that the series $\ds \sum_{n \mathop = 1}^\infty a_n$ is itself convergent. | ;Proof by Counterexample
Consider the harmonic sequence:
:$\sequence {a_n} = \sequence {\dfrac 1 n}$
from Sequence of Reciprocals is Null Sequence:
:$\ds \lim_{n \mathop \to \infty} \dfrac 1 n = 0$
However, from Harmonic Series is Divergent, $\ds s = \sum_{n \mathop = 1}^\infty \dfrac 1 n$ is divergent.
Hence the resul... | Let $\sequence {a_n}$ be a [[Definition:Sequence|sequence]] in any of the [[Definition:Standard Number Field|standard number fields]] [[Definition:Rational Number|$\Q$]], [[Definition:Real Number|$\R$]], or [[Definition:Complex Number|$\C$]].
Suppose that $\sequence {a_n}$ [[Definition:Convergent Sequence|converges]] ... | ;[[Proof by Counterexample]]
Consider the [[Definition:Harmonic Sequence|harmonic sequence]]:
:$\sequence {a_n} = \sequence {\dfrac 1 n}$
from [[Sequence of Reciprocals is Null Sequence]]:
:$\ds \lim_{n \mathop \to \infty} \dfrac 1 n = 0$
However, from [[Harmonic Series is Divergent]], $\ds s = \sum_{n \mathop = 1}^... | Series whose Terms Converge to Zero is not necessarily Convergent | https://proofwiki.org/wiki/Series_whose_Terms_Converge_to_Zero_is_not_necessarily_Convergent | https://proofwiki.org/wiki/Series_whose_Terms_Converge_to_Zero_is_not_necessarily_Convergent | [
"Divergent Series",
"Convergent Series",
"Convergent Sequences"
] | [
"Definition:Sequence",
"Definition:Standard Number Field",
"Definition:Rational Number",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Convergent Sequence",
"Definition:Zero (Number)",
"Definition:Series",
"Definition:Convergent Series"
] | [
"Proof by Counterexample",
"Definition:Harmonic Sequence",
"Sequence of Powers of Reciprocals is Null Sequence/Corollary",
"Harmonic Series is Divergent",
"Definition:Divergent Series"
] |
proofwiki-22879 | Closed Subspace of Separable Banach Space Linearly Isomorphic to Space of Zero-Limit Sequences is Topologically Complemented | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a separable Banach space over $\GF$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
Let $E$ be a closed vector subspace of $X$ that is linearly isomorphic to $\map {C_0} \GF$.
Then $E$ is topologic... | Let $T : E \to \map {c_0} \GF$ be a linear isomorphism.
Then $T^{-1} : \map {c_0} \GF \to E$ is a linear isomorphism.
From Sobczyk's Theorem, there exists a bounded linear transformation $\widetilde T : X \to \map {c_0} \GF$ extending $T$.
Let $P = T^{-1} \widetilde T : X \to E$.
Then $P$ is a bounded linear transforma... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Separable Space|separable]] [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Zero-Limit Sequences|space of zero-limit sequences]].
Let $E... | Let $T : E \to \map {c_0} \GF$ be a [[Definition:Linear Isomorphism|linear isomorphism]].
Then $T^{-1} : \map {c_0} \GF \to E$ is a [[Definition:Linear Isomorphism|linear isomorphism]].
From [[Sobczyk's Theorem]], there exists a [[Definition:Bounded Linear Transformation|bounded linear transformation]] $\widetilde T ... | Closed Subspace of Separable Banach Space Linearly Isomorphic to Space of Zero-Limit Sequences is Topologically Complemented | https://proofwiki.org/wiki/Closed_Subspace_of_Separable_Banach_Space_Linearly_Isomorphic_to_Space_of_Zero-Limit_Sequences_is_Topologically_Complemented | https://proofwiki.org/wiki/Closed_Subspace_of_Separable_Banach_Space_Linearly_Isomorphic_to_Space_of_Zero-Limit_Sequences_is_Topologically_Complemented | [
"Space of Zero-Limit Sequences"
] | [
"Definition:Separable Space",
"Definition:Banach Space",
"Definition:Space of Zero-Limit Sequences",
"Definition:Closed Set",
"Definition:Vector Subspace",
"Definition:Linear Isomorphism",
"Definition:Topologically Complemented Subspace of Banach Space"
] | [
"Definition:Linear Isomorphism",
"Definition:Linear Isomorphism",
"Sobczyk's Theorem",
"Definition:Bounded Linear Transformation",
"Definition:Extension of Mapping",
"Definition:Bounded Linear Transformation",
"Definition:Extension of Mapping",
"Definition:Bounded Linear Transformation",
"Definition... |
proofwiki-22880 | Injective Closed Subspace of Banach Space is Topologically Complemented | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $E$ be a closed vector subspace of $X$ that is injective.
Then $E$ is topologically complemented in $X$. | Define $I : E \to E$ by:
:$I x = x$ for each $x \in E$.
Since $E$ is an injective Banach space, there exists a bounded linear transformation $P : X \to E$ that extends $I$.
Then, for each $x \in E$, we have $P x = I x = x$, and hence $P^2 x = P x$.
So $P$ is also a projection.
Hence $E$ is topologically complemented in... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $E$ be a [[Definition:Closed Set|closed]] [[Definition:Vector Subspace|vector subspace]] of $X$ that is [[Definition:Injective Banach Space|injective]].
Then $E$ is [[Definition:Topolo... | Define $I : E \to E$ by:
:$I x = x$ for each $x \in E$.
Since $E$ is an [[Definition:Injective Banach Space|injective Banach space]], there exists a [[Definition:Bounded Linear Transformation|bounded linear transformation]] $P : X \to E$ that [[Definition:Extension of Mapping|extends]] $I$.
Then, for each $x \in E$, ... | Injective Closed Subspace of Banach Space is Topologically Complemented | https://proofwiki.org/wiki/Injective_Closed_Subspace_of_Banach_Space_is_Topologically_Complemented | https://proofwiki.org/wiki/Injective_Closed_Subspace_of_Banach_Space_is_Topologically_Complemented | [
"Topologically Complemented Subspaces of Banach Spaces",
"Injective Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Closed Set",
"Definition:Vector Subspace",
"Definition:Injective Banach Space",
"Definition:Topologically Complemented Subspace of Banach Space"
] | [
"Definition:Injective Banach Space",
"Definition:Bounded Linear Transformation",
"Definition:Extension of Mapping",
"Definition:Projection (Vector Spaces)",
"Definition:Topologically Complemented Subspace of Banach Space",
"Category:Topologically Complemented Subspaces of Banach Spaces",
"Category:Injec... |
proofwiki-22881 | Space of Zero-Limit Sequences is not Injective | Let $\GF \in \set {\R, \C}$.
Let $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ be the space of zero-limit sequences.
Then $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ is not injective. | Let $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$ be the space of bounded sequences.
From Space of Zero-Limit Sequences with Supremum Norm forms Banach Space, $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ is indeed a Banach space.
From Subspace of Complete Metric Space is Closed iff Complete, ... | 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]].
Then $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ is not [[Definition:Injective Banach Space|injective]]. | Let $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Bounded Sequences|space of bounded sequences]].
From [[Space of Zero-Limit Sequences with Supremum Norm forms Banach Space]], $\struct {\map {c_0} \GF, \norm {\, \cdot \,}_\infty}$ is indeed a [[Definition:Banach Space|Bana... | Space of Zero-Limit Sequences is not Injective | https://proofwiki.org/wiki/Space_of_Zero-Limit_Sequences_is_not_Injective | https://proofwiki.org/wiki/Space_of_Zero-Limit_Sequences_is_not_Injective | [
"Injective Banach Spaces",
"Space of Zero-Limit Sequences"
] | [
"Definition:Space of Zero-Limit Sequences",
"Definition:Injective Banach Space"
] | [
"Definition:Space of Bounded Sequences",
"Space of Zero-Limit Sequences with Supremum Norm forms Banach Space",
"Definition:Banach Space",
"Subspace of Complete Metric Space is Closed iff Complete",
"Definition:Closed Set",
"Definition:Injective Banach Space",
"Injective Closed Subspace of Banach Space ... |
proofwiki-22882 | Characterization of Isomorphism in Loc* | Let $\mathbf{Loc_*}$ denote the category of locales with localic mappings.
Let $\mathbf{Frm}$ denote the category of frames.
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 of $\mathbf{Loc_*}$.
Let $\loweradjoint f : L_2 \to L_1$ be the lower ... | === Statement $(1)$ Iff Statement (2) ===
{{:Characterization of Isomorphism in Loc*/Isomorphism iff Lower Adjoint is Frame Isomorphism}}{{qed|lemma}} | Let $\mathbf{Loc_*}$ denote the [[Definition:Category of Locales with Localic Mappings|category of locales with localic mappings]].
Let $\mathbf{Frm}$ denote the [[Definition:Category of Frames|category of frames]].
Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Locale (Latti... | === [[Characterization of Isomorphism in Loc*/Isomorphism iff Lower Adjoint is Frame Isomorphism|Statement $(1)$ Iff Statement (2)]] ===
{{:Characterization of Isomorphism in Loc*/Isomorphism iff Lower Adjoint is Frame Isomorphism}}{{qed|lemma}} | Characterization of Isomorphism in Loc* | https://proofwiki.org/wiki/Characterization_of_Isomorphism_in_Loc* | https://proofwiki.org/wiki/Characterization_of_Isomorphism_in_Loc* | [
"Locales",
"Characterization of Isomorphism in Loc*"
] | [
"Definition:Category of Locales with Localic Mappings",
"Definition:Category of Frames",
"Definition:Locale (Lattice Theory)",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Galois Connection/Lower Adjoint",
"Definition:Isomorphism",
"Definition:Frame Isomorphism",
"Definition:Order... | [
"Characterization of Isomorphism in Loc*/Isomorphism iff Lower Adjoint is Frame Isomorphism"
] |
proofwiki-22883 | Injective Separable Banach Space is Isometrically Isomorphic to Topologically Complemented Subspace of Space of Bounded Sequences | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a injective separable Banach space over $\GF$.
Let $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$ be the space of bounded sequences.
Then there exists a topologically complemented subspace $E$ of $\map {\ell_\infty} \GF$ that is is... | From Separable Normed Vector Space Isometrically Isomorphic to Linear Subspace of Space of Bounded Sequences, there exists a linear isometry $T : X \to \map {\ell_\infty} \GF$ such that $T : X \to T \sqbrk X$ is an isometric isomorphism.
Hence $T \sqbrk X$ is a closed vector subspace of $\map {\ell_\infty} \GF$.
We sho... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Injective Banach Space|injective]] [[Definition:Separable Space|separable]] [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\struct {\map {\ell_\infty} \GF, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Bound... | From [[Separable Normed Vector Space Isometrically Isomorphic to Linear Subspace of Space of Bounded Sequences]], there exists a [[Definition:Linear Isometry|linear isometry]] $T : X \to \map {\ell_\infty} \GF$ such that $T : X \to T \sqbrk X$ is an [[Definition:Isometric Isomorphism/Normed Vector Space|isometric isomo... | Injective Separable Banach Space is Isometrically Isomorphic to Topologically Complemented Subspace of Space of Bounded Sequences | https://proofwiki.org/wiki/Injective_Separable_Banach_Space_is_Isometrically_Isomorphic_to_Topologically_Complemented_Subspace_of_Space_of_Bounded_Sequences | https://proofwiki.org/wiki/Injective_Separable_Banach_Space_is_Isometrically_Isomorphic_to_Topologically_Complemented_Subspace_of_Space_of_Bounded_Sequences | [
"Injective Banach Spaces",
"Separable Spaces",
"Space of Bounded Sequences"
] | [
"Definition:Injective Banach Space",
"Definition:Separable Space",
"Definition:Banach Space",
"Definition:Space of Bounded Sequences",
"Definition:Topologically Complemented Subspace of Banach Space",
"Definition:Isometric Isomorphism/Normed Vector Space"
] | [
"Separable Normed Vector Space Isometrically Isomorphic to Linear Subspace of Space of Bounded Sequences",
"Definition:Linear Isometry",
"Definition:Isometric Isomorphism/Normed Vector Space",
"Definition:Closed Set",
"Definition:Vector Subspace",
"Definition:Topologically Complemented Subspace of Banach ... |
proofwiki-22884 | Characterization of Isomorphism in Loc*/Isomorphism iff Lower Adjoint is Frame Isomorphism | Let $\mathbf{Loc_*}$ denote the category of locales with localic mappings.
Let $\mathbf{Frm}$ denote the category of frames.
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 of $\mathbf{Loc_*}$.
Let $\loweradjoint f : L_2 \to L_1$ be the lower ... | ==== Necessary Condition ====
Let $f$ be an isomorphism of $\mathbf{Loc_*}$.
By definition of isomorphism:
:there exists a localic mapping $g: L_2 \to L_1$ of $\mathbf{Loc_*}$:
::$g \circ f = \operatorname{id}_{L_1}$
:and
::$f \circ g = \operatorname{id}_{L_2}$
By definition of localic mapping:
:$g$ has a lower adjoint... | Let $\mathbf{Loc_*}$ denote the [[Definition:Category of Locales with Localic Mappings|category of locales with localic mappings]].
Let $\mathbf{Frm}$ denote the [[Definition:Category of Frames|category of frames]].
Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Locale (Latti... | ==== Necessary Condition ====
Let $f$ be an [[Definition:Isomorphism (Category Theory)|isomorphism]] of $\mathbf{Loc_*}$.
By definition of [[Definition:Isomorphism (Category Theory)|isomorphism]]:
:there exists a [[Definition:Localic Mapping|localic mapping]] $g: L_2 \to L_1$ of $\mathbf{Loc_*}$:
::$g \circ f = \ope... | Characterization of Isomorphism in Loc*/Isomorphism iff Lower Adjoint is Frame Isomorphism | https://proofwiki.org/wiki/Characterization_of_Isomorphism_in_Loc*/Isomorphism_iff_Lower_Adjoint_is_Frame_Isomorphism | https://proofwiki.org/wiki/Characterization_of_Isomorphism_in_Loc*/Isomorphism_iff_Lower_Adjoint_is_Frame_Isomorphism | [
"Characterization of Isomorphism in Loc*"
] | [
"Definition:Category of Locales with Localic Mappings",
"Definition:Category of Frames",
"Definition:Locale (Lattice Theory)",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Galois Connection/Lower Adjoint",
"Definition:Isomorphism",
"Definition:Frame Isomorphism"
] | [
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Galois Connection/Lower Adjoint",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definit... |
proofwiki-22885 | Characterization of Isomorphism in Loc*/Order Isomorphism iff Lower Adjoint is Order Isomorphism | Let $\mathbf{Loc_*}$ denote the category of locales with localic mappings.
Let $\mathbf{Frm}$ denote the category of frames.
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 of $\mathbf{Loc_*}$.
Let $\loweradjoint f : L_2 \to L_1$ be the lower ... | ==== Necessary Condition ====
Let $\loweradjoint f$ be an order isomorphism.
From Inverse of Order Isomorphism is Order Isomorphism:
:$\paren{\loweradjoint f}^{-1}$ is an order isomorphism
From Order Isomorphism forms Galois Connection:
:$\tuple{\paren{\loweradjoint f}^{-1}, \loweradjoint f}$ is a Galois connection
By ... | Let $\mathbf{Loc_*}$ denote the [[Definition:Category of Locales with Localic Mappings|category of locales with localic mappings]].
Let $\mathbf{Frm}$ denote the [[Definition:Category of Frames|category of frames]].
Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Locale (Latti... | ==== Necessary Condition ====
Let $\loweradjoint f$ be an [[Definition:Order Isomorphism|order isomorphism]].
From [[Inverse of Order Isomorphism is Order Isomorphism]]:
:$\paren{\loweradjoint f}^{-1}$ is an [[Definition:Order Isomorphism|order isomorphism]]
From [[Order Isomorphism forms Galois Connection]]:
:$\t... | Characterization of Isomorphism in Loc*/Order Isomorphism iff Lower Adjoint is Order Isomorphism | https://proofwiki.org/wiki/Characterization_of_Isomorphism_in_Loc*/Order_Isomorphism_iff_Lower_Adjoint_is_Order_Isomorphism | https://proofwiki.org/wiki/Characterization_of_Isomorphism_in_Loc*/Order_Isomorphism_iff_Lower_Adjoint_is_Order_Isomorphism | [
"Characterization of Isomorphism in Loc*"
] | [
"Definition:Category of Locales with Localic Mappings",
"Definition:Category of Frames",
"Definition:Locale (Lattice Theory)",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Galois Connection/Lower Adjoint",
"Definition:Order Isomorphism",
"Definition:Order Isomorphism"
] | [
"Definition:Order Isomorphism",
"Inverse of Order Isomorphism is Order Isomorphism",
"Definition:Order Isomorphism",
"Order Isomorphism forms Galois Connection",
"Definition:Galois Connection",
"Definition:Continuous Map (Locale)/Localic Mapping",
"Definition:Galois Connection",
"Galois Connection is ... |
proofwiki-22886 | Sequence in Metric Space with Summable Difference between Consecutive Terms is Cauchy Sequence | Let $\struct {X, d}$ be a metric space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $\struct {X, d}$ such that:
:$\ds \sum_{k \mathop = 1}^\infty \map d {x_k, x_{k + 1} } < \infty$
Then $\sequence {x_n}_{n \mathop \in \N}$ is a Cauchy sequence. | Since:
:$\ds \sum_{k \mathop = 1}^\infty \map d {x_k, x_{k + 1} } < \infty$
we have that:
:$\ds \sequence {\sum_{k \mathop = 1}^n \map d {x_k, x_{k + 1} } }_{n \mathop \in \N}$ is a Cauchy sequence.
Let $\epsilon > 0$.
Then there exists $N \in \N$ such that:
:$\ds \size {\sum_{k \mathop = 1}^{n - 1} \map d {x_k, x_{k +... | 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 $\struct {X, d}$ such that:
:$\ds \sum_{k \mathop = 1}^\infty \map d {x_k, x_{k + 1} } < \infty$
Then $\sequence {x_n}_{n \mathop \in \N}$ is a [[Definition:Cauchy Seq... | Since:
:$\ds \sum_{k \mathop = 1}^\infty \map d {x_k, x_{k + 1} } < \infty$
we have that:
:$\ds \sequence {\sum_{k \mathop = 1}^n \map d {x_k, x_{k + 1} } }_{n \mathop \in \N}$ is a [[Definition:Cauchy Sequence|Cauchy sequence]].
Let $\epsilon > 0$.
Then there exists $N \in \N$ such that:
:$\ds \size {\sum_{k \mathop... | Sequence in Metric Space with Summable Difference between Consecutive Terms is Cauchy Sequence | https://proofwiki.org/wiki/Sequence_in_Metric_Space_with_Summable_Difference_between_Consecutive_Terms_is_Cauchy_Sequence | https://proofwiki.org/wiki/Sequence_in_Metric_Space_with_Summable_Difference_between_Consecutive_Terms_is_Cauchy_Sequence | [
"Sequence in Metric Space with Summable Difference between Consecutive Terms is Cauchy Sequence",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Cauchy Sequence"
] | [
"Definition:Cauchy Sequence",
"Definition:Cauchy Sequence",
"Category:Sequence in Metric Space with Summable Difference between Consecutive Terms is Cauchy Sequence",
"Category:Metric Spaces"
] |
proofwiki-22887 | Convolution Integral is Commutative | Let $f$ and $g$ be real functions which are integrable.
Let $\map f t * \map g t$ denote the convolution integral of $f$ and $g$.
Then:
:$\map f t * \map g t = \map g t * \map f t$
That is, the convolution integral is commutative. | {{begin-eqn}}
{{eqn | l = \map f t * \map g t
| r = \int_{-\infty}^\infty \map f u \map g {t - u} \rd u
| c = {{Defof|Convolution Integral}}
}}
{{eqn | r = -\int_\infty^{-\infty} \map f {t - v} \map g v \rd v
| c = Integration by Substitution with $\begin {cases} v & = & t - u \\ \d v & = & -\d u \end... | Let $f$ and $g$ be [[Definition:Real Function|real functions]] which are [[Definition:Integrable Function|integrable]].
Let $\map f t * \map g t$ denote the [[Definition:Convolution Integral|convolution integral]] of $f$ and $g$.
Then:
:$\map f t * \map g t = \map g t * \map f t$
That is, the [[Definition:Convolutio... | {{begin-eqn}}
{{eqn | l = \map f t * \map g t
| r = \int_{-\infty}^\infty \map f u \map g {t - u} \rd u
| c = {{Defof|Convolution Integral}}
}}
{{eqn | r = -\int_\infty^{-\infty} \map f {t - v} \map g v \rd v
| c = [[Integration by Substitution]] with $\begin {cases} v & = & t - u \\ \d v & = & -\d u ... | Convolution Integral is Commutative | https://proofwiki.org/wiki/Convolution_Integral_is_Commutative | https://proofwiki.org/wiki/Convolution_Integral_is_Commutative | [
"Convolution Integrals",
"Examples of Commutative Operations"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Convolution Integral",
"Definition:Convolution Integral",
"Definition:Commutative/Operation"
] | [
"Integration by Substitution",
"Reversal of Limits of Definite Integral"
] |
proofwiki-22888 | Ekeland's Variational Principle | Let $\struct {X, d}$ be a complete metric space.
Let $\phi : X \to \R \cup \set \infty$ be a lower semicontinuous function that is bounded below.
Then for each $\epsilon > 0$, there exists $x_0 \in X$ such that:
:$\map \phi x \ge \map \phi {x_0} - \epsilon \map d {x, x_0}$ for each $x \in X$. | Take $M \in \N$ such that:
:$\map \phi x \ge M$ for each $x \in X$.
Then, for any $y \in X$, we have:
:$\map \phi x + \map d {x, y} \ge M$
Hence, for each $y \in X$:
:$\inf \set {\map \phi x + \map d {x, y} : x \in X}$ is finite.
Fix any $y_1 \in X$.
We construct a sequence $\sequence {y_n}_{n \mathop \in \N}$ induct... | Let $\struct {X, d}$ be a [[Definition:Complete Metric Space|complete metric space]].
Let $\phi : X \to \R \cup \set \infty$ be a [[Definition:Lower Semicontinuous|lower semicontinuous function]] that is [[Definition:Bounded Below Mapping|bounded below]].
Then for each $\epsilon > 0$, there exists $x_0 \in X$ such t... | Take $M \in \N$ such that:
:$\map \phi x \ge M$ for each $x \in X$.
Then, for any $y \in X$, we have:
:$\map \phi x + \map d {x, y} \ge M$
Hence, for each $y \in X$:
:$\inf \set {\map \phi x + \map d {x, y} : x \in X}$ is [[Definition:Finite Extended Real Number|finite]].
Fix any $y_1 \in X$.
We construct a [[Def... | Ekeland's Variational Principle | https://proofwiki.org/wiki/Ekeland's_Variational_Principle | https://proofwiki.org/wiki/Ekeland's_Variational_Principle | [
"Lower Semicontinuity",
"Complete Metric Spaces"
] | [
"Definition:Complete Metric Space",
"Definition:Lower Semicontinuous",
"Definition:Bounded Below Mapping"
] | [
"Definition:Finite Extended Real Number",
"Definition:Sequence",
"Definition:Decreasing/Sequence",
"Monotone Convergence Theorem (Real Analysis)/Decreasing Sequence",
"Definition:Convergent Sequence",
"Combination Theorem for Sequences/Real/Sum Rule",
"Definition:Convergent Sequence",
"Telescoping Ser... |
proofwiki-22889 | Conway Circle is Circle | Let $\triangle ABC$ be a triangle.
Let $\KK$ be the Conway circle of $ABC$.
Then $\KK$ is indeed a circle. | {{ProofWanted|Easy enough to get from the internet if anyone wants to post this up, it's a fun little thing}} | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $\KK$ be the [[Definition:Conway Circle|Conway circle]] of $ABC$.
Then $\KK$ is indeed a [[Definition:Circle|circle]]. | {{ProofWanted|Easy enough to get from the internet if anyone wants to post this up, it's a fun little thing}} | Conway Circle is Circle | https://proofwiki.org/wiki/Conway_Circle_is_Circle | https://proofwiki.org/wiki/Conway_Circle_is_Circle | [
"Conway Circles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Conway Circle",
"Definition:Circle"
] | [] |
proofwiki-22890 | Center of Conway Circle is Incenter | Let $\triangle ABC$ be a triangle.
Let $\KK$ be the Conway circle of $ABC$.
Then the center of $\KK$ coincides with the incenter of $\triangle ABC$. | {{ProofWanted|As with Conway Circle is Circle the proof of one leads to the other}} | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $\KK$ be the [[Definition:Conway Circle|Conway circle]] of $ABC$.
Then the [[Definition:Center of Circle|center]] of $\KK$ coincides with the [[Definition:Incenter of Triangle|incenter]] of $\triangle ABC$. | {{ProofWanted|As with [[Conway Circle is Circle]] the proof of one leads to the other}} | Center of Conway Circle is Incenter | https://proofwiki.org/wiki/Center_of_Conway_Circle_is_Incenter | https://proofwiki.org/wiki/Center_of_Conway_Circle_is_Incenter | [
"Conway Circles",
"Incenters"
] | [
"Definition:Triangle (Geometry)",
"Definition:Conway Circle",
"Definition:Circle/Center",
"Definition:Incircle of Triangle/Incenter"
] | [
"Conway Circle is Circle"
] |
proofwiki-22891 | Function is Convex iff Strict Epigraph is Convex | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Consider $X \times \GF$ as the direct product of $X$ and $\GF$.
Let $f : X \to \R$ be a function.
Let $\map {\operatorname {epi}_S} f$ be the strict epigraph of $f$.
Then $f$ is convex {{iff}} $\map {\operatorname {epi}_S} f$ is convex in $X \times \GF$... | === Necessary Condition ===
Suppose that $f$ is convex.
Let $t \in \openint 0 1$.
Let $\tuple {x_1, \alpha_1}, \tuple {x_2, \alpha_2} \in \map {\operatorname {epi} } f$.
Then:
:$\map f {x_1} < \alpha_1$
and:
:$\map f {x_2} < \alpha_2$
We therefore have, since $t > 0$:
:$t \map f {x_1} + \paren {1 - t} \map f {x_2} < ... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Consider $X \times \GF$ as the [[Definition:Direct Product of Vector Spaces|direct product]] of $X$ and $\GF$.
Let $f : X \to \R$ be a [[Definition:Function|function]].
Let $\map {\operatorname {epi}_S} f$ be the [[Defini... | === Necessary Condition ===
Suppose that $f$ is [[Definition:Convex Real Function|convex]].
Let $t \in \openint 0 1$.
Let $\tuple {x_1, \alpha_1}, \tuple {x_2, \alpha_2} \in \map {\operatorname {epi} } f$.
Then:
:$\map f {x_1} < \alpha_1$
and:
:$\map f {x_2} < \alpha_2$
We therefore have, since $t > 0$:
:$t \map... | Function is Convex iff Strict Epigraph is Convex | https://proofwiki.org/wiki/Function_is_Convex_iff_Strict_Epigraph_is_Convex | https://proofwiki.org/wiki/Function_is_Convex_iff_Strict_Epigraph_is_Convex | [
"Convex Real Functions",
"Convex Sets (Vector Spaces)",
"Epigraphs"
] | [
"Definition:Vector Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Function",
"Definition:Epigraph/Strict",
"Definition:Convex Real Function",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Convex Real Function",
"Definition:Convex Real Function",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Real Function"
] |
proofwiki-22892 | Function is Concave iff Hypograph is Convex | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Consider $X \times \GF$ as the direct product of $X$ and $\GF$.
Let $f : X \to \R$ be a function.
Let $\map {\operatorname {hypo} } f$ be the hypograph of $f$.
Then $f$ is concave {{iff}} $\map {\operatorname {hypo} } f$ is convex in $X \times \GF$. | === Necessary Condition ===
Suppose that $f$ is concave.
Let $t \in \openint 0 1$.
Let $\tuple {x_1, \alpha_1}, \tuple {x_2, \alpha_2} \in \map {\operatorname {epi} } f$.
Then:
:$\map f {x_1} \ge \alpha_1$
and:
:$\map f {x_2} \ge \alpha_2$
We therefore have, since $t > 0$:
:$t \map f {x_1} + \paren {1 - t} \map f {x_... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Consider $X \times \GF$ as the [[Definition:Direct Product of Vector Spaces|direct product]] of $X$ and $\GF$.
Let $f : X \to \R$ be a [[Definition:Function|function]].
Let $\map {\operatorname {hypo} } f$ be the [[Defini... | === Necessary Condition ===
Suppose that $f$ is [[Definition:Concave Real Function|concave]].
Let $t \in \openint 0 1$.
Let $\tuple {x_1, \alpha_1}, \tuple {x_2, \alpha_2} \in \map {\operatorname {epi} } f$.
Then:
:$\map f {x_1} \ge \alpha_1$
and:
:$\map f {x_2} \ge \alpha_2$
We therefore have, since $t > 0$:
:$... | Function is Concave iff Hypograph is Convex | https://proofwiki.org/wiki/Function_is_Concave_iff_Hypograph_is_Convex | https://proofwiki.org/wiki/Function_is_Concave_iff_Hypograph_is_Convex | [
"Concave Real Functions",
"Convex Sets (Vector Spaces)",
"Hypographs"
] | [
"Definition:Vector Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Function",
"Definition:Hypograph",
"Definition:Concave Real Function",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Concave Real Function",
"Definition:Concave Real Function",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Concave Real Function"
] |
proofwiki-22893 | Function is Concave iff Strict Hypograph is Convex | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Consider $X \times \GF$ as the direct product of $X$ and $\GF$.
Let $f : X \to \R$ be a function.
Let $\map {\operatorname {hypo} } f$ be the hypograph of $f$.
Then $f$ is convex {{iff}} $\map {\operatorname {hypo} } f$ is convex in $X \times \GF$. | === Necessary Condition ===
Suppose that $f$ is convex.
Let $t \in \openint 0 1$.
Let $\tuple {x_1, \alpha_1}, \tuple {x_2, \alpha_2} \in \map {\operatorname {epi} } f$.
Then:
:$\map f {x_1} > \alpha_1$
and:
:$\map f {x_2} > \alpha_2$
We therefore have, since $t > 0$:
:$t \map f {x_1} + \paren {1 - t} \map f {x_2} > ... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Consider $X \times \GF$ as the [[Definition:Direct Product of Vector Spaces|direct product]] of $X$ and $\GF$.
Let $f : X \to \R$ be a [[Definition:Function|function]].
Let $\map {\operatorname {hypo} } f$ be the [[Defini... | === Necessary Condition ===
Suppose that $f$ is [[Definition:Convex Real Function|convex]].
Let $t \in \openint 0 1$.
Let $\tuple {x_1, \alpha_1}, \tuple {x_2, \alpha_2} \in \map {\operatorname {epi} } f$.
Then:
:$\map f {x_1} > \alpha_1$
and:
:$\map f {x_2} > \alpha_2$
We therefore have, since $t > 0$:
:$t \map... | Function is Concave iff Strict Hypograph is Convex | https://proofwiki.org/wiki/Function_is_Concave_iff_Strict_Hypograph_is_Convex | https://proofwiki.org/wiki/Function_is_Concave_iff_Strict_Hypograph_is_Convex | [
"Concave Real Functions",
"Convex Sets (Vector Spaces)",
"Hypographs"
] | [
"Definition:Vector Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Function",
"Definition:Hypograph",
"Definition:Concave Real Function",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Convex Real Function",
"Definition:Concave Real Function",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Concave Real Function"
] |
proofwiki-22894 | Real Constant Function is Convex and Concave | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $c \in \R$.
Let $f : X \to \R$ be a real function by:
:$\map f x = c$
Then $f$ is convex and concave. | Let $x, y \in X$ and $t \in \openint 0 1$.
We have:
:$\map f {t x + \paren {1 - t} y} = c$
We also have:
{{begin-eqn}}
{{eqn | l = t \map f x + \paren {1 - t} \map f y
| r = t c + \paren {1 - t} c
}}
{{eqn | r = c
}}
{{eqn | r = \map f {t x + \paren {1 - t} y}
}}
{{end-eqn}}
Hence we have:
:$\map f {t x + \paren {1 -... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $c \in \R$.
Let $f : X \to \R$ be a [[Definition:Real Function|real function]] by:
:$\map f x = c$
Then $f$ is [[Definition:Convex Real Function/Vector Space|convex]] and [[Definition:Concave Real Function/Vector Spa... | Let $x, y \in X$ and $t \in \openint 0 1$.
We have:
:$\map f {t x + \paren {1 - t} y} = c$
We also have:
{{begin-eqn}}
{{eqn | l = t \map f x + \paren {1 - t} \map f y
| r = t c + \paren {1 - t} c
}}
{{eqn | r = c
}}
{{eqn | r = \map f {t x + \paren {1 - t} y}
}}
{{end-eqn}}
Hence we have:
:$\map f {t x + \paren {... | Real Constant Function is Convex and Concave | https://proofwiki.org/wiki/Real_Constant_Function_is_Convex_and_Concave | https://proofwiki.org/wiki/Real_Constant_Function_is_Convex_and_Concave | [
"Convex Real Functions",
"Concave Real Functions"
] | [
"Definition:Vector Space",
"Definition:Real Function",
"Definition:Convex Real Function/Vector Space",
"Definition:Concave Real Function/Vector Space"
] | [
"Definition:Convex Real Function/Vector Space",
"Definition:Concave Real Function/Vector Space",
"Category:Convex Real Functions",
"Category:Concave Real Functions"
] |
proofwiki-22895 | Non-Negative Multiple of Convex Real Function is Convex | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $f : X \to \R$ be a convex real function.
Let $\lambda \ge 0$.
Then $\lambda f$ is convex. | Let $x, y \in X$ and $t \in \openint 0 1$.
Then:
:$\map f {t x + \paren {1 - t} y} \le t \map f x + \paren {1 - t} \map f y$
Multiplying through by $\lambda$ we have:
:$\lambda \map f {t x + \paren {1 - t} y} \le t \lambda \map f x + \paren {1 - t} \lambda \map f y$
Hence $\lambda f$ is convex.
{{qed}}
Category:Convex ... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $f : X \to \R$ be a [[Definition:Convex Real Function/Vector Space|convex real function]].
Let $\lambda \ge 0$.
Then $\lambda f$ is [[Definition:Convex Real Function/Vector Space|convex]]. | Let $x, y \in X$ and $t \in \openint 0 1$.
Then:
:$\map f {t x + \paren {1 - t} y} \le t \map f x + \paren {1 - t} \map f y$
Multiplying through by $\lambda$ we have:
:$\lambda \map f {t x + \paren {1 - t} y} \le t \lambda \map f x + \paren {1 - t} \lambda \map f y$
Hence $\lambda f$ is [[Definition:Convex Real Func... | Non-Negative Multiple of Convex Real Function is Convex | https://proofwiki.org/wiki/Non-Negative_Multiple_of_Convex_Real_Function_is_Convex | https://proofwiki.org/wiki/Non-Negative_Multiple_of_Convex_Real_Function_is_Convex | [
"Convex Real Functions"
] | [
"Definition:Vector Space",
"Definition:Convex Real Function/Vector Space",
"Definition:Convex Real Function/Vector Space"
] | [
"Definition:Convex Real Function/Vector Space",
"Category:Convex Real Functions"
] |
proofwiki-22896 | Sum of Convex Real Functions is Convex | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $f, g : X \to \R$ be a convex real function.
Then $f + g$ is a convex real function. | Let $x, y \in X$ and $t \in \openint 0 1$.
Since $f$ is convex, we have:
:$\map f {t x + \paren {1 - t} y} \le t \map f x + \paren {1 - t} \map f y$
Since $g$ is convex, we have:
:$\map g {t x + \paren {1 - t} y} \le t \map g x + \paren {1 - t} \map g y$
Hence:
{{begin-eqn}}
{{eqn | l = \map {\paren {f + g} } {t x + \... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $f, g : X \to \R$ be a [[Definition:Convex Real Function/Vector Space|convex real function]].
Then $f + g$ is a [[Definition:Convex Real Function/Vector Space|convex real function]]. | Let $x, y \in X$ and $t \in \openint 0 1$.
Since $f$ is [[Definition:Convex Real Function/Vector Space|convex]], we have:
:$\map f {t x + \paren {1 - t} y} \le t \map f x + \paren {1 - t} \map f y$
Since $g$ is [[Definition:Convex Real Function/Vector Space|convex]], we have:
:$\map g {t x + \paren {1 - t} y} \le t ... | Sum of Convex Real Functions is Convex | https://proofwiki.org/wiki/Sum_of_Convex_Real_Functions_is_Convex | https://proofwiki.org/wiki/Sum_of_Convex_Real_Functions_is_Convex | [
"Convex Real Functions"
] | [
"Definition:Vector Space",
"Definition:Convex Real Function/Vector Space",
"Definition:Convex Real Function/Vector Space"
] | [
"Definition:Convex Real Function/Vector Space",
"Definition:Convex Real Function/Vector Space",
"Definition:Convex Real Function/Vector Space",
"Category:Convex Real Functions"
] |
proofwiki-22897 | Correlation need not imply Causation | Let $X$ and $Y$ be random variables which show correlation between them.
Then it is not necessarily the case that they linked by cause and effect. | Take the example of car ownership and alcohol sales.
{{:Correlation/Examples/Car Ownership and Alcohol Sales}}{{qed}} | Let $X$ and $Y$ be [[Definition:Random Variable|random variables]] which show [[Definition:Correlation|correlation]] between them.
Then it is not necessarily the case that they linked by cause and effect. | Take the example of [[Correlation/Examples/Car Ownership and Alcohol Sales|car ownership and alcohol sales]].
{{:Correlation/Examples/Car Ownership and Alcohol Sales}}{{qed}} | Correlation need not imply Causation | https://proofwiki.org/wiki/Correlation_need_not_imply_Causation | https://proofwiki.org/wiki/Correlation_need_not_imply_Causation | [
"Correlation"
] | [
"Definition:Random Variable",
"Definition:Correlation"
] | [
"Correlation/Examples/Car Ownership and Alcohol Sales"
] |
proofwiki-22898 | Hom Bifunctor With Left Functor is Covariant Functor | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$ and $\mathbf D$ be locally small categories.
Let $L : \mathbf D \to \mathbf C$ be a covariant functor.
Let $\map {\operatorname{Hom}_{\mathbf C} } {L-, -} : \mathbf D^{\text{op} } \times \mathbf C \to \mathbf {Set}$ denote the hom bifunctor with left functor.... | === Object Functor is Well-defined ===
By definition of functor:
:for each object $D$ in $\mathbf D$:
::$LD$ is an object in $\mathbf C$
By definition of dual category:
:for each object $D^\text{op}$ in $\mathbf D^\text{op}$:
::$LD$ is an object in $\mathbf C$
By definition of locally small category:
:for each object... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$ and $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $L : \mathbf D \to \mathbf C$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\map {\operatorname{Hom}_{\mathbf C} } {L-, -... | === Object Functor is Well-defined ===
By definition of [[Definition:Covariant Functor|functor]]:
:for each [[Definition:Object (Category Theory)|object]] $D$ in $\mathbf D$:
::$LD$ is an [[Definition:Object (Category Theory)|object]] in $\mathbf C$
By definition of [[Definition:Dual Category|dual category]]:
:for e... | Hom Bifunctor With Left Functor is Covariant Functor | https://proofwiki.org/wiki/Hom_Bifunctor_With_Left_Functor_is_Covariant_Functor | https://proofwiki.org/wiki/Hom_Bifunctor_With_Left_Functor_is_Covariant_Functor | [
"Bifunctors"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Hom Bifunctor With Left Functor",
"Definition:Functor/Covariant"
] | [
"Definition:Functor/Covariant",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Dual Category",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Locally Small Category",
"Definition:Object (Category Theory)",
"Definit... |
proofwiki-22899 | Hom Bifunctor With Right Functor is Covariant Functor | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$ and $\mathbf D$ be locally small categories.
Let $R : \mathbf C \to \mathbf D$ be a covariant functor.
Let $\map {\operatorname{Hom}_{\mathbf D} } {-, R-} : \mathbf D^{\text{op} } \times \mathbf C \to \mathbf {Set}$ denote the hom bifunctor with right functor... | === Object Functor is Well-defined ===
By definition of functor:
:for each object $C$ in $\mathbf C$:
::$RC$ is an object in $\mathbf D$
By definition of locally small category:
:for each object $\tuple{D^\text{op}, C}$ in $\mathbf D^\text{op} \times \mathbf C$:
::$\map {\operatorname{Hom}_{\mathbf D} } {D, RC}$ is a... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$ and $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $R : \mathbf C \to \mathbf D$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\map {\operatorname{Hom}_{\mathbf D} } {-, R-... | === Object Functor is Well-defined ===
By definition of [[Definition:Covariant Functor|functor]]:
:for each [[Definition:Object (Category Theory)|object]] $C$ in $\mathbf C$:
::$RC$ is an [[Definition:Object (Category Theory)|object]] in $\mathbf D$
By definition of [[Definition:Locally Small Category|locally small... | Hom Bifunctor With Right Functor is Covariant Functor | https://proofwiki.org/wiki/Hom_Bifunctor_With_Right_Functor_is_Covariant_Functor | https://proofwiki.org/wiki/Hom_Bifunctor_With_Right_Functor_is_Covariant_Functor | [
"Bifunctors"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Hom Bifunctor With Right Functor",
"Definition:Functor/Covariant"
] | [
"Definition:Functor/Covariant",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Locally Small Category",
"Definition:Object (Category Theory)",
"Definition:Set",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Hom Bi... |
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