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 |
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proofwiki-20400 | Quotient Mapping is Bounded in Normed Quotient Vector Space | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a normed vector space over $\Bbb F$.
Let $N$ be a closed linear subspace of $X$.
Let $\struct {X/N, \norm {\, \cdot \,}_{X/N} }$ be the normed quotient vector space associated with quotient vector space $X/N$.
Let $\pi : X \to X/N$ be the quotient mapping associated with $X/N... | From Quotient Mapping is Linear Transformation:
:$\pi$ is a linear transformation.
Let $x \in X$.
Note that from the definition of quotient norm, we have:
:$\ds \norm {\map \pi x}_{X/N} = \inf_{z \in N} \norm {x - z}_X$
Note that since $0 \in N$, we have that:
:$\norm x_X \in \set {\norm {x - z}_X : z \in N}$
so th... | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $N$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $X$.
Let $\struct {X/N, \norm {\, \cdot \,}_{X/N} }$ be the [[Definition:Normed Quotient Vector Space|normed quotient vector... | From [[Quotient Mapping is Linear Transformation]]:
:$\pi$ is a [[Definition:Linear Transformation|linear transformation]].
Let $x \in X$.
Note that from the definition of [[Definition:Quotient Norm|quotient norm]], we have:
:$\ds \norm {\map \pi x}_{X/N} = \inf_{z \in N} \norm {x - z}_X$
Note that since $0 \in ... | Quotient Mapping is Bounded in Normed Quotient Vector Space | https://proofwiki.org/wiki/Quotient_Mapping_is_Bounded_in_Normed_Quotient_Vector_Space | https://proofwiki.org/wiki/Quotient_Mapping_is_Bounded_in_Normed_Quotient_Vector_Space | [
"Quotient Mappings",
"Normed Quotient Vector Spaces",
"Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Linear Subspace",
"Definition:Normed Quotient Vector Space",
"Definition:Quotient Vector Space",
"Definition:Quotient Mapping",
"Definition:Bounded Linear Transformation"
] | [
"Quotient Mapping is Linear Transformation",
"Definition:Linear Transformation",
"Definition:Quotient Norm",
"Definition:Infimum of Set/Real Numbers",
"Definition:Bounded Linear Transformation",
"Category:Quotient Mappings",
"Category:Normed Quotient Vector Spaces",
"Category:Bounded Linear Transforma... |
proofwiki-20401 | Bing's Metrization Theorem | Let $T = \struct {S, \tau}$ be a topological space.
Then:
:$T$ is metrizable {{iff}} $T$ is regular and has a $\sigma$-discrete basis | {{proof wanted}}
{{Namedfor|R.H. Bing|cat = Bing}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then:
:$T$ is [[Definition:Metrizable Space|metrizable]] {{iff}} $T$ is [[Definition:Regular Space|regular]] and has a [[Definition:Sigma-Discrete Basis|$\sigma$-discrete basis]] | {{proof wanted}}
{{Namedfor|R.H. Bing|cat = Bing}} | Bing's Metrization Theorem | https://proofwiki.org/wiki/Bing's_Metrization_Theorem | https://proofwiki.org/wiki/Bing's_Metrization_Theorem | [
"Metrization Theorems",
"Regular Spaces",
"Metrizable Spaces"
] | [
"Definition:Topological Space",
"Definition:Metrizable Space",
"Definition:Regular Space",
"Definition:Sigma-Discrete Basis"
] | [] |
proofwiki-20402 | Cantor's Diagonal Argument/Corollary | Let $S$ be a set such that $\card S > 1$, that is, such that $S$ is not a singleton.
Let $\mathbb G$ be the set of all mappings from the integers $\Z$ to $S$:
:$\mathbb G = \set {f: \Z \to S}$
Then $\mathbb G$ is uncountably infinite. | Let $\mathbb F$ be the set of all mappings from the natural numbers $\N$ to $S$:
:$\mathbb F = \set {f: \N \to S}$
From Cantor's Diagonal Argument, $\mathbb F$ is uncountably infinite.
Let $s \in S$ be an arbitrary distinguished element of $S$.
Let $\mathbb H$ be the set of mappings $h: \Z \to S$ defined as:
:$\forall ... | Let $S$ be a [[Definition:Set|set]] such that $\card S > 1$, that is, such that $S$ is not a [[Definition:Singleton|singleton]].
Let $\mathbb G$ be the [[Definition:Set|set]] of all [[Definition:Mapping|mappings]] from the [[Definition:Integer|integers]] $\Z$ to $S$:
:$\mathbb G = \set {f: \Z \to S}$
Then $\mathbb G$... | Let $\mathbb F$ be the [[Definition:Set|set]] of all [[Definition:Mapping|mappings]] from the [[Definition:Natural Numbers|natural numbers]] $\N$ to $S$:
:$\mathbb F = \set {f: \N \to S}$
From [[Cantor's Diagonal Argument]], $\mathbb F$ is [[Definition:Uncountable Set|uncountably infinite]].
Let $s \in S$ be an arbit... | Cantor's Diagonal Argument/Corollary | https://proofwiki.org/wiki/Cantor's_Diagonal_Argument/Corollary | https://proofwiki.org/wiki/Cantor's_Diagonal_Argument/Corollary | [
"Cantor's Diagonal Argument"
] | [
"Definition:Set",
"Definition:Singleton",
"Definition:Set",
"Definition:Mapping",
"Definition:Integer",
"Definition:Uncountable/Set"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Natural Numbers",
"Cantor's Diagonal Argument",
"Definition:Uncountable/Set",
"Definition:Distinguished Element",
"Definition:Set",
"Definition:Mapping",
"Definition:Set Equivalence",
"Definition:Uncountable/Set",
"Definition:Uncountable/Set",
... |
proofwiki-20403 | Quotient Mapping Maps Unit Open Ball in Normed Vector Space to Unit Open Ball in Normed Quotient Vector Space | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $N$ be a closed linear subspace of $X$.
Let $\struct {X/N, \norm {\, \cdot \,} }$ be the normed quotient vector space associated with the quotient vector space $X/N$.
Let $B_X$ be the unit open ball in $\struct {X, \norm {\, \cdot \,} }$.
Let $B_{X/N... | From Quotient Mapping is Bounded in Normed Quotient Vector Space, we have:
:$\norm {\map \pi x}_{X/N} \le \norm x$
So if $x \in B_X$, we have $\norm x < 1$ and hence:
:$\norm {\map \pi x}_{X/N} < 1$
So $\map \pi x \in B_{X/N}$.
So we have:
:$\map \pi {B_X} \subseteq B_{X/N}$
Conversely, let $\mathbf x \in B_{X/N}$ ... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $N$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $X$.
Let $\struct {X/N, \norm {\, \cdot \,} }$ be the [[Definition:Normed Quotient Vector Space|normed quotient vector space]] associated wi... | From [[Quotient Mapping is Bounded in Normed Quotient Vector Space]], we have:
:$\norm {\map \pi x}_{X/N} \le \norm x$
So if $x \in B_X$, we have $\norm x < 1$ and hence:
:$\norm {\map \pi x}_{X/N} < 1$
So $\map \pi x \in B_{X/N}$.
So we have:
:$\map \pi {B_X} \subseteq B_{X/N}$
Conversely, let $\mathbf x \i... | Quotient Mapping Maps Unit Open Ball in Normed Vector Space to Unit Open Ball in Normed Quotient Vector Space | https://proofwiki.org/wiki/Quotient_Mapping_Maps_Unit_Open_Ball_in_Normed_Vector_Space_to_Unit_Open_Ball_in_Normed_Quotient_Vector_Space | https://proofwiki.org/wiki/Quotient_Mapping_Maps_Unit_Open_Ball_in_Normed_Vector_Space_to_Unit_Open_Ball_in_Normed_Quotient_Vector_Space | [
"Quotient Mappings",
"Normed Quotient Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Linear Subspace",
"Definition:Normed Quotient Vector Space",
"Definition:Quotient Vector Space",
"Definition:Unit Open Ball",
"Definition:Unit Open Ball",
"Definition:Quotient Mapping"
] | [
"Quotient Mapping is Bounded in Normed Quotient Vector Space",
"Kernel of Quotient Mapping",
"Quotient Mapping is Linear Transformation",
"Category:Quotient Mappings",
"Category:Normed Quotient Vector Spaces"
] |
proofwiki-20404 | Separable Normed Vector Space Isometrically Isomorphic to Linear Subspace of Space of Bounded Sequences | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a separable normed vector space over $\Bbb F$.
Let $\struct {\map {\ell^\infty} {\Bbb F}, \norm {\, \cdot \,}_\infty}$ be the normed vector space of bounded sequences.
Then there exists a linear subspace $Y$ of $\map {\ell^\infty} {\Bbb F}$ suc... | Let $\mathcal S = \set {x_n : n \in \N}$ be a countable everywhere dense subset of $X$.
By Existence of Support Functional, for each $n \in \N$ there exists $f_n \in X^\ast$ such that $\norm {f_n}_{X^\ast} = 1$ and $\map {f_n} {x_n} = \norm {x_n}$.
Then, for each $x \in X$, we have:
:$\cmod {\map {f_n} x} \le \norm x... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Separable Space|separable]] [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $\struct {\map {\ell^\infty} {\Bbb F}, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Space of Bounded Sequences/Normed Vec... | Let $\mathcal S = \set {x_n : n \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere dense]] [[Definition:Subset|subset]] of $X$.
By [[Existence of Support Functional]], for each $n \in \N$ there exists $f_n \in X^\ast$ such that $\norm {f_n}_{X^\ast} = 1$ and $\map {f_n} {x_n... | Separable Normed Vector Space Isometrically Isomorphic to Linear Subspace of Space of Bounded Sequences | https://proofwiki.org/wiki/Separable_Normed_Vector_Space_Isometrically_Isomorphic_to_Linear_Subspace_of_Space_of_Bounded_Sequences | https://proofwiki.org/wiki/Separable_Normed_Vector_Space_Isometrically_Isomorphic_to_Linear_Subspace_of_Space_of_Bounded_Sequences | [
"Normed Vector Spaces",
"Space of Bounded Sequences",
"Isometric Isomorphisms (Normed Vector Spaces)",
"Separable Spaces",
"Normed Vector Spaces",
"Space of Bounded Sequences"
] | [
"Definition:Separable Space",
"Definition:Normed Vector Space",
"Definition:Space of Bounded Sequences/Normed Vector Space",
"Definition:Linear Subspace",
"Definition:Isometric Isomorphism/Normed Vector Space"
] | [
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Definition:Subset",
"Existence of Support Functional",
"Supremum Operator Norm as Universal Upper Bound",
"Image of Vector Subspace under Linear Transformation is Vector Subspace",
"Definition:Vector Subspace",
"Definition:Linear Isometry",
... |
proofwiki-20405 | Characterization of Complete Normed Quotient Vector Spaces | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $N$ be a closed linear subspace of $X$.
Let $\struct {X/N, \norm {\, \cdot \,}_{X/N} }$ be the normed quotient vector space associated with the quotient vector space $X/N$.
Let $\norm {\, \cdot \,}_N$ be the norm on $N$ given by restricting the nor... | Let $\pi : X \to X/N$ be the quotient mapping associated with $X/N$. | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $N$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $X$.
Let $\struct {X/N, \norm {\, \cdot \,}_{X/N} }$ be the [[Definition:Normed Quotient Vector Space|normed quotient vector space]] assoc... | Let $\pi : X \to X/N$ be the [[Definition:Quotient Mapping|quotient mapping]] associated with $X/N$. | Characterization of Complete Normed Quotient Vector Spaces | https://proofwiki.org/wiki/Characterization_of_Complete_Normed_Quotient_Vector_Spaces | https://proofwiki.org/wiki/Characterization_of_Complete_Normed_Quotient_Vector_Spaces | [
"Banach Spaces",
"Normed Quotient Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Linear Subspace",
"Definition:Normed Quotient Vector Space",
"Definition:Quotient Vector Space",
"Definition:Norm/Vector Space",
"Definition:Norm/Vector Space",
"Definition:Banach Space",
"Definition:Banach Space"
] | [
"Definition:Quotient Mapping"
] |
proofwiki-20406 | Bounded Linear Transformation preserves Cauchy Sequences | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces.
Let $T : X \to Y$ be a bounded linear transformation.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $X$.
Then $\sequence {T x_n}_{n \mathop \in \N}$ is a Cauchy sequence in $Y$. | Since $T$ is a bounded linear transformation, there exists $M > 0$ such that:
:$\norm {T x}_Y \le M \norm x_X$
for all $x \in X$.
So, since $T$ is linear, we have:
:$\norm {T x_n - T x_m}_Y \le M \norm {x_n - x_m}_X$
Let $\epsilon > 0$.
Since $\sequence {x_n}_{n \mathop \in \N}$ is Cauchy, there exists $N \in \N$ suc... | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $T : X \to Y$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Cauchy Sequence in... | Since $T$ is a [[Definition:Bounded Linear Transformation|bounded linear transformation]], there exists $M > 0$ such that:
:$\norm {T x}_Y \le M \norm x_X$
for all $x \in X$.
So, since $T$ is [[Definition:Linear Transformation|linear]], we have:
:$\norm {T x_n - T x_m}_Y \le M \norm {x_n - x_m}_X$
Let $\epsilon ... | Bounded Linear Transformation preserves Cauchy Sequences | https://proofwiki.org/wiki/Bounded_Linear_Transformation_preserves_Cauchy_Sequences | https://proofwiki.org/wiki/Bounded_Linear_Transformation_preserves_Cauchy_Sequences | [
"Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Bounded Linear Transformation",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space"
] | [
"Definition:Bounded Linear Transformation",
"Definition:Linear Transformation",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space",
"Category:Bounded Linear Transformations"
] |
proofwiki-20407 | Number of Atoms in Observable Universe | The number of atoms in the observable universe is approximately $10^{80}$. | According to our current observations and understanding of the {{WP|Universe}}, the age of the universe is estimated to be $13.787$ billion years.
That means the light from the most distant galaxies has traveled $13.787$ billion years to reach us.
The speed of light is $299 \, 792 \, 458 \ \text {m s}^{-1}$
This implie... | The number of [[Definition:Atom (Physics)|atoms]] in the [[Definition:Observable Universe|observable universe]] is approximately $10^{80}$. | According to our current observations and understanding of the {{WP|Universe}}, the [[Definition:Age (Time)|age]] of the [[Definition:Physical Universe|universe]] is estimated to be $13.787$ [[Definition:Billion (Short Scale) |billion]] [[Definition:Year|years]].
That means the [[Definition:Light (Radiation)|light]] f... | Number of Atoms in Observable Universe | https://proofwiki.org/wiki/Number_of_Atoms_in_Observable_Universe | https://proofwiki.org/wiki/Number_of_Atoms_in_Observable_Universe | [
"Atoms"
] | [
"Definition:Atom (Physics)",
"Definition:Observable Universe"
] | [
"Definition:Age (Time)",
"Definition:Physical Universe",
"Definition:Billion (Short Scale) ",
"Definition:Time/Unit/Year",
"Definition:Light (Radiation)",
"Definition:Galaxy",
"Definition:Billion (Short Scale) ",
"Definition:Time/Unit/Year",
"Definition:Speed of Light",
"Definition:Sphere/Geometry... |
proofwiki-20408 | Metrization of Regular Second Countable Space | Let $T = \struct {S, \tau}$ be a $T_1$ space.
{{TFAE}}
:$(1): \quad T$ is regular and second-countable
:$(2): \quad T$ is homeomorphic to a metric subspace of the Hilbert cube $I^\omega$
:$(3): \quad T$ is metrizable and separable | === Condition $(1)$ implies Condition $(2)$ ===
Follows immediately from Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:T1 Space|$T_1$ space]].
{{TFAE}}
:$(1): \quad T$ is [[Definition:Regular|regular]] and [[Definition:Second-Countable Space|second-countable]]
:$(2): \quad T$ is [[Definition:Homeomorphism|homeomorphic]] to a [[Definition:Metric Subspace|metric subspace]] of the [[Definiti... | === Condition $(1)$ implies Condition $(2)$ ===
Follows immediately from [[Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube]].
{{qed|lemma}} | Metrization of Regular Second Countable Space | https://proofwiki.org/wiki/Metrization_of_Regular_Second_Countable_Space | https://proofwiki.org/wiki/Metrization_of_Regular_Second_Countable_Space | [
"Metrizable Spaces",
"Regular Spaces",
"Separable Spaces",
"Second-Countable Spaces",
"T1 Spaces"
] | [
"Definition:T1 Space",
"Definition:Regular",
"Definition:Second-Countable Space",
"Definition:Homeomorphism",
"Definition:Metric Subspace",
"Definition:Hilbert Cube",
"Definition:Metrizable",
"Definition:Separable"
] | [
"Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube"
] |
proofwiki-20409 | Equivalence of Definitions of Sine and Cosine | In the following, $\theta$ understood to take values in $\hointr 0 {2 \pi}$. | Consider the following vector-valued function $\mathbf f : \R \to {\closedint {-1} 1}^2$:
:$\map {\mathbf f} t = \tuple {\cos t, \sin t}$
where $\cos t$ and $\sin t$ are defined analytically.
Then, for any $t$ the distance to the origin is:
{{begin-eqn}}
{{eqn | l = \norm {\map {\mathbf f} t - \bszero}
| r = \nor... | In the following, $\theta$ understood to take values in $\hointr 0 {2 \pi}$. | Consider the following [[Definition:Vector-Valued Function|vector-valued function]] $\mathbf f : \R \to {\closedint {-1} 1}^2$:
:$\map {\mathbf f} t = \tuple {\cos t, \sin t}$
where $\cos t$ and $\sin t$ are [[Definition:Real Cosine Function|defined]] [[Definition:Real Sine Function|analytically]].
Then, for any $t$ ... | Equivalence of Definitions of Sine and Cosine | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Sine_and_Cosine | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Sine_and_Cosine | [
"Sine Function",
"Cosine Function",
"Definition Equivalences"
] | [] | [
"Definition:Vector-Valued Function",
"Definition:Cosine/Real Function",
"Definition:Sine/Real Function",
"Definition:Distance between Points",
"Definition:Coordinate System/Origin",
"Sum of Squares of Sine and Cosine",
"Definition:Unit Circle",
"Definition:Arc Length",
"Arc Length for Parametric Equ... |
proofwiki-20410 | Construction of Transitive Closure of Relation | Let $\RR$ be a relation.
Let $\RR^+$ be the relation which is constructed from $\RR$ as follows:
:$(1): \quad$ If $\tuple {a, b} \in \RR$, then $\tuple {a, b} \in \RR^+$
:$(2): \quad$ If $\tuple {a, b} \in \RR^+$ and $\tuple {b, c} \in \RR$, then $\tuple {a, c} \in \RR^+$
:$(3): \quad$ Nothing is in $\RR^+$ unless it s... | Let $\tuple {x, y} \in \R^+$ from rules $(1)$ and $(2)$.
Then either:
:$\tuple {x, y}$ belongs there because $\RR \subseteq \RR^+$
or:
:$\tuple {x, y}$ belongs there, because if it were not then $\RR^+$ would not be transitive.
It remains to be shown that $\RR^+$ is in fact transitive.
{{finish|"Easy inductive proof", ... | Let $\RR$ be a [[Definition:Relation|relation]].
Let $\RR^+$ be the [[Definition:Relation|relation]] which is constructed from $\RR$ as follows:
:$(1): \quad$ If $\tuple {a, b} \in \RR$, then $\tuple {a, b} \in \RR^+$
:$(2): \quad$ If $\tuple {a, b} \in \RR^+$ and $\tuple {b, c} \in \RR$, then $\tuple {a, c} \in \RR... | Let $\tuple {x, y} \in \R^+$ from rules $(1)$ and $(2)$.
Then either:
:$\tuple {x, y}$ belongs there because $\RR \subseteq \RR^+$
or:
:$\tuple {x, y}$ belongs there, because if it were not then $\RR^+$ would not be [[Definition:Transitive Relation|transitive]].
It remains to be shown that $\RR^+$ is in fact [[Defin... | Construction of Transitive Closure of Relation | https://proofwiki.org/wiki/Construction_of_Transitive_Closure_of_Relation | https://proofwiki.org/wiki/Construction_of_Transitive_Closure_of_Relation | [
"Transitive Closures"
] | [
"Definition:Relation",
"Definition:Relation",
"Definition:Transitive Closure of Relation"
] | [
"Definition:Transitive Relation",
"Definition:Transitive Relation",
"Definition:Smallest Set by Set Inclusion",
"Definition:Transitive Relation",
"Definition:Subset"
] |
proofwiki-20411 | Sub-Basis for Initial Topology in terms of Sub-Bases of Target Spaces | Let $X$ be a set.
Let $I$ be an indexing set.
Let $\family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces indexed by $I$.
For each $i \in I$, let $S_i$ be a synthetic basis for $\struct {Y_i, \tau_i}$.
Let $\family {f_i: X \to Y_i}_{i \mathop \in I}$ be an indexed family of mapp... | Note that by the definition of the initial topology, $\tau$ is generated by the synthetic sub-basis:
:$\SS' = \set {f_i^{-1} \sqbrk U : i \in I, \, U \in \tau_i}$
Since $S_i \subseteq \tau_i$ for each $i \in I$, we have:
:$\SS \subseteq \SS'$
and hence:
:$\map \tau \SS \subseteq \map \tau {\SS'} = \tau$
where $\tau... | Let $X$ be a [[Definition:Set|set]].
Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] [[Definition:Indexing Set|indexed]] by $I$.
For each $i \in I... | Note that by the definition of the [[Definition:Initial Topology|initial topology]], $\tau$ is [[Definition:Topology Generated by Synthetic Sub-Basis|generated]] by the [[Definition:Synthetic Sub-Basis|synthetic sub-basis]]:
:$\SS' = \set {f_i^{-1} \sqbrk U : i \in I, \, U \in \tau_i}$
Since $S_i \subseteq \tau_i$ ... | Sub-Basis for Initial Topology in terms of Sub-Bases of Target Spaces | https://proofwiki.org/wiki/Sub-Basis_for_Initial_Topology_in_terms_of_Sub-Bases_of_Target_Spaces | https://proofwiki.org/wiki/Sub-Basis_for_Initial_Topology_in_terms_of_Sub-Bases_of_Target_Spaces | [
"Initial Topologies",
"Initial Topology",
"Initial Topology",
"Topological Bases"
] | [
"Definition:Set",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Sub-Basis/Synthetic Sub-Basis",
"Definition:Indexing Set/Family",
"Definition:Mapping",
"Definition:Indexing Set",
"Definition:Initial Topology",
... | [
"Definition:Initial Topology",
"Definition:Topology Generated by Synthetic Sub-Basis",
"Definition:Sub-Basis/Synthetic Sub-Basis",
"Definition:Topology Generated by Synthetic Sub-Basis",
"Definition:Sub-Basis/Synthetic Sub-Basis",
"Definition:Indexing Set",
"Preimage of Union under Mapping",
"Preimage... |
proofwiki-20412 | Sequence of Natural Powers of Right Shift Operator in 2-Sequence Space does not Converge in Strong Operator Topology | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the $2$-sequence normed vector space.
Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the continuous linear transformation space.
Let $R \in \map {CL} {\ell^2}$ be the right shift operator over $\ell^2$.
Let $\sequence {R^n}_{n \mathop \in \N}$ be a sequence.
L... | Let $\mathbf e_1 = \tuple {1, 0, \ldots} \in \ell^2$
Then $R^n \mathbf e_1 = \tuple {\underbrace {\ldots, 0}_{n \text{ terms} }, 1, 0, \ldots}$
So:
:$\forall n \in \N : \norm {R^n \mathbf e_1}_2 = 1$
Therefore:
:$\ds \lim_{n \mathop \to \infty} \norm {R^n \mathbf e_1}_2 = 1$
Hence, $\sequence {R^n}_{n \mathop \in \N}$ ... | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the [[P-Sequence Space with P-Norm forms Normed Vector Space|$2$-sequence normed vector space]].
Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $R \in \map ... | Let $\mathbf e_1 = \tuple {1, 0, \ldots} \in \ell^2$
Then $R^n \mathbf e_1 = \tuple {\underbrace {\ldots, 0}_{n \text{ terms} }, 1, 0, \ldots}$
So:
:$\forall n \in \N : \norm {R^n \mathbf e_1}_2 = 1$
Therefore:
:$\ds \lim_{n \mathop \to \infty} \norm {R^n \mathbf e_1}_2 = 1$
Hence, $\sequence {R^n}_{n \mathop \in... | Sequence of Natural Powers of Right Shift Operator in 2-Sequence Space does not Converge in Strong Operator Topology | https://proofwiki.org/wiki/Sequence_of_Natural_Powers_of_Right_Shift_Operator_in_2-Sequence_Space_does_not_Converge_in_Strong_Operator_Topology | https://proofwiki.org/wiki/Sequence_of_Natural_Powers_of_Right_Shift_Operator_in_2-Sequence_Space_does_not_Converge_in_Strong_Operator_Topology | [
"Convergence",
"Operator Theory"
] | [
"P-Sequence Space with P-Norm forms Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Right Shift Operator",
"Definition:Sequence",
"Definition:Zero Mapping/Vector Space",
"Definition:Convergent Sequence in Weak Operator Topology",
"Definition:Strong Operator Topolog... | [
"Definition:Convergent Sequence in Strong Operator Topology",
"Definition:Strong Operator Topology"
] |
proofwiki-20413 | Pointwise Minimum of Metric and Positive Real Number is Topologically Equivalent Metric | Let $\struct {X, d}$ be a metric space.
Let $c > 0$ be a real number.
For each $x, y \in X$, define:
:$\map {d'} {x, y} = \min \set {\map d {x, y}, c}$
Then $d'$ is a metric that is topologically equivalent to $d$. | === {{Metric-space-axiom|1|nolink}} ===
Suppose that $x, y \in X$ are such that:
:$\map {d'} {x, y} = 0$
Since $c > 0$, this implies that:
:$\map d {x, y} = 0$
Since $d$ is a metric, we have $x = y$ by {{Metric-space-axiom|1}} for $d$.
Hence {{Metric-space-axiom|1}} is fulfilled.
{{qed|lemma}} | Let $\struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $c > 0$ be a [[Definition:Real Number|real number]].
For each $x, y \in X$, define:
:$\map {d'} {x, y} = \min \set {\map d {x, y}, c}$
Then $d'$ is a [[Definition:Metric|metric]] that is [[Definition:Topologically Equivalent Metrics|topolog... | === {{Metric-space-axiom|1|nolink}} ===
Suppose that $x, y \in X$ are such that:
:$\map {d'} {x, y} = 0$
Since $c > 0$, this implies that:
:$\map d {x, y} = 0$
Since $d$ is a [[Definition:Metric|metric]], we have $x = y$ by {{Metric-space-axiom|1}} for $d$.
Hence {{Metric-space-axiom|1}} is fulfilled.
{{qed|le... | Pointwise Minimum of Metric and Positive Real Number is Topologically Equivalent Metric | https://proofwiki.org/wiki/Pointwise_Minimum_of_Metric_and_Positive_Real_Number_is_Topologically_Equivalent_Metric | https://proofwiki.org/wiki/Pointwise_Minimum_of_Metric_and_Positive_Real_Number_is_Topologically_Equivalent_Metric | [
"Metric Spaces",
"Topologically Equivalent Metrics"
] | [
"Definition:Metric Space",
"Definition:Real Number",
"Definition:Metric Space/Metric",
"Definition:Topologically Equivalent Metrics"
] | [
"Definition:Metric Space/Metric",
"Definition:Metric Space/Metric"
] |
proofwiki-20414 | Composition of Continuous Linear Transformations is Continuous Linear Transformation | Let $K$ be a field.
Let $\struct {X, \norm {\, \cdot \,}_X}$, $\struct {Y, \norm {\, \cdot \,}_Y}$, $\struct{Z, \norm {\, \cdot \,}_Z}$ be normed vector spaces over $K$.
$\map {CL} {X, Y}$ be the continuous linear transformation space.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Let $S \circ T : X \to Z$ b... | === Linearity ===
Follows from Composition of Linear Transformations is Linear Transformation. | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\struct {X, \norm {\, \cdot \,}_X}$, $\struct {Y, \norm {\, \cdot \,}_Y}$, $\struct{Z, \norm {\, \cdot \,}_Z}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $K$.
$\map {CL} {X, Y}$ be the [[Definition:Continuous Linear Transformation S... | === Linearity ===
Follows from [[Composition of Linear Transformations is Linear Transformation]]. | Composition of Continuous Linear Transformations is Continuous Linear Transformation | https://proofwiki.org/wiki/Composition_of_Continuous_Linear_Transformations_is_Continuous_Linear_Transformation | https://proofwiki.org/wiki/Composition_of_Continuous_Linear_Transformations_is_Continuous_Linear_Transformation | [
"Functional Analysis",
"Mapping Theory",
"Continuity",
"Linear Transformations"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Supremum Operator Norm",
"Definition:Composition of Mappings"
] | [
"Composition of Linear Transformations is Linear Transformation"
] |
proofwiki-20415 | Characterization of Fineness of Topology in terms of Topological Bases | Let $X$ be a set.
Let $\tau$ and $\tau'$ be topologies on $X$.
Let $\BB \subseteq \tau$ be a basis for $\tau$ and $\BB' \subseteq \tau'$ be a basis for $\tau'$.
{{TFAE}}
:$(1): \quad$ $\tau'$ is finer than $\tau$
:$(2): \quad$ for each $x \in X$ and each $B \in \BB$ containing $x$, there exists $B' \in \BB'$ such that... | === $(1)$ implies $(2)$ ===
Let $x \in X$ and $B \in \BB$ contain $x$.
Since $\tau'$ is finer than $\tau$, we have that $B \in \tau'$.
Since $\BB'$ is a basis for $\tau'$, there exists a subset $\set {B'_\alpha : \alpha \in I} \subseteq \BB'$ such that:
:$\ds B = \bigcup_{\alpha \mathop \in I} B'_\alpha$
from Open Se... | Let $X$ be a [[Definition:Set|set]].
Let $\tau$ and $\tau'$ be [[Definition:Topology|topologies]] on $X$.
Let $\BB \subseteq \tau$ be a [[Definition:Basis (Topology)|basis]] for $\tau$ and $\BB' \subseteq \tau'$ be a [[Definition:Basis (Topology)|basis]] for $\tau'$.
{{TFAE}}
:$(1): \quad$ $\tau'$ is [[Definition... | === $(1)$ implies $(2)$ ===
Let $x \in X$ and $B \in \BB$ contain $x$.
Since $\tau'$ is [[Definition:Finer Topology|finer]] than $\tau$, we have that $B \in \tau'$.
Since $\BB'$ is a [[Definition:Basis (Topology)|basis]] for $\tau'$, there exists a [[Definition:Subset|subset]] $\set {B'_\alpha : \alpha \in I} \subs... | Characterization of Fineness of Topology in terms of Topological Bases | https://proofwiki.org/wiki/Characterization_of_Fineness_of_Topology_in_terms_of_Topological_Bases | https://proofwiki.org/wiki/Characterization_of_Fineness_of_Topology_in_terms_of_Topological_Bases | [
"Topological Bases"
] | [
"Definition:Set",
"Definition:Topology",
"Definition:Basis (Topology)",
"Definition:Basis (Topology)",
"Definition:Finer Topology"
] | [
"Definition:Finer Topology",
"Definition:Basis (Topology)",
"Definition:Subset",
"Open Set is Union of Elements of Basis",
"Set is Subset of Union",
"Definition:Basis (Topology)",
"Definition:Subset",
"Open Set is Union of Elements of Basis"
] |
proofwiki-20416 | Modulus of Linear Functional on Vector Space is Seminorm | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $f : X \to \GF$ be a linear functional.
Define $p_f : X \to \R_{\ge 0}$ by:
:$\map {p_f} x = \cmod {\map f x}$
for each $x \in X$.
Then $p_f$ is a seminorm. | === Proof of {{SeminormAxiom|2}} ===
For each $\lambda \in \GF$ and $x \in X$, we have:
{{begin-eqn}}
{{eqn | l = \map {p_f} {\lambda x}
| r = \cmod {\map f {\lambda x} }
}}
{{eqn | r = \cmod {\lambda \map f x}
| c = since $f$ is linear
}}
{{eqn | r = \cmod \lambda \cmod {\map f x}
}}
{{eqn | r = \cmod \lambda \ma... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $f : X \to \GF$ be a [[Definition:Linear Functional|linear functional]].
Define $p_f : X \to \R_{\ge 0}$ by:
:$\map {p_f} x = \cmod {\map f x}$
for each $x \in X$.
Then $p_f$ is a [[Definition:Seminorm|seminorm]... | === Proof of {{SeminormAxiom|2}} ===
For each $\lambda \in \GF$ and $x \in X$, we have:
{{begin-eqn}}
{{eqn | l = \map {p_f} {\lambda x}
| r = \cmod {\map f {\lambda x} }
}}
{{eqn | r = \cmod {\lambda \map f x}
| c = since $f$ is [[Definition:Linear Functional|linear]]
}}
{{eqn | r = \cmod \lambda \cmod {\map f ... | Modulus of Linear Functional on Vector Space is Seminorm | https://proofwiki.org/wiki/Modulus_of_Linear_Functional_on_Vector_Space_is_Seminorm | https://proofwiki.org/wiki/Modulus_of_Linear_Functional_on_Vector_Space_is_Seminorm | [
"Seminorms"
] | [
"Definition:Vector Space",
"Definition:Linear Functional",
"Definition:Seminorm"
] | [
"Definition:Linear Functional",
"Definition:Linear Functional"
] |
proofwiki-20417 | Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $F$ be a set of linear functionals on $X$ that form a vector space over $\GF$.
That is, for each $\lambda, \mu \in \GF$ and $f, g \in F$, we have:
:$\lambda f + \mu g \in F$
Let $\tau$ be the initial topology on $X$ generated by $F$.
For each $f ... | Let $\tau'$ be the standard topology on the locally convex space.
From definition, a sub-basis for $\tau'$ is given by:
:$\SS' = \set {\map {B_{p_f} } {\epsilon, x} : f \in F, \, \epsilon > 0, \, x \in X}$
where:
:$\map {B_{p_f} } {\epsilon, x} = \set {y \in X : \map {p_f} {y - x} < \epsilon}$
For each $u \in \GF$, l... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $F$ be a set of [[Definition:Linear Functional|linear functionals]] on $X$ that form a [[Definition:Vector Space|vector space]] over $\GF$.
That is, for each $\lambda, \mu \in \GF$ and $f, g \in F$, we have:
:$\lam... | Let $\tau'$ be the [[Definition:Locally Convex Space/Standard Topology|standard topology]] on the [[Definition:Locally Convex Space|locally convex space]].
From definition, a [[Definition:Sub-Basis|sub-basis]] for $\tau'$ is given by:
:$\SS' = \set {\map {B_{p_f} } {\epsilon, x} : f \in F, \, \epsilon > 0, \, x \in ... | Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex | https://proofwiki.org/wiki/Initial_Topology_on_Vector_Space_Generated_by_Linear_Functionals_is_Locally_Convex | https://proofwiki.org/wiki/Initial_Topology_on_Vector_Space_Generated_by_Linear_Functionals_is_Locally_Convex | [
"Locally Convex Spaces",
"Linear Functionals",
"Initial Topology"
] | [
"Definition:Vector Space",
"Definition:Linear Functional",
"Definition:Vector Space",
"Definition:Initial Topology",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Locally Convex Space"
] | [
"Definition:Locally Convex Space/Standard Topology",
"Definition:Locally Convex Space",
"Definition:Sub-Basis",
"Open Balls form Basis for Open Sets of Metric Space",
"Definition:Basis (Topology)",
"Sub-Basis for Initial Topology in terms of Sub-Bases of Target Spaces",
"Definition:Sub-Basis",
"Linear... |
proofwiki-20418 | Linearly Independent Set is Contained in some Basis/Infinite Dimensional Case | Let $K$ be a field.
Let $E$ be a vector space over $K$.
Let $H$ be a linearly independent subset of $E$.
There exists a basis $B$ for $E$ such that $H \subseteq B$. | Let:
:$\SS = \set {L \supseteq H : L \subseteq E \text { is linearly independent} }$
We have $H \in \SS$, so certainly $\SS \ne \O$.
With view to apply Zorn's Lemma, we show that every non-empty $\subseteq$-chain in $\SS$ has an upper bound.
Let $\CC$ be a non-empty $\subseteq$-chain in $\SS$.
Let:
:$\ds C = \bigcup ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $E$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $H$ be a [[Definition:Linearly Independent Set|linearly independent subset]] of $E$.
There exists a [[Definition:Basis of Vector Space|basis]] $B$ for $E$ such that $H \subseteq B$. | Let:
:$\SS = \set {L \supseteq H : L \subseteq E \text { is linearly independent} }$
We have $H \in \SS$, so certainly $\SS \ne \O$.
With view to apply [[Zorn's Lemma]], we show that every [[Definition:Non-Empty Set|non-empty]] [[Definition:Chain (Order Theory)|$\subseteq$-chain]] in $\SS$ has an [[Definition:Upper... | Linearly Independent Set is Contained in some Basis/Infinite Dimensional Case | https://proofwiki.org/wiki/Linearly_Independent_Set_is_Contained_in_some_Basis/Infinite_Dimensional_Case | https://proofwiki.org/wiki/Linearly_Independent_Set_is_Contained_in_some_Basis/Infinite_Dimensional_Case | [
"Linearly Independent Set is Contained in some Basis"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Linearly Independent/Set",
"Definition:Basis of Vector Space"
] | [
"Zorn's Lemma",
"Definition:Non-Empty Set",
"Definition:Chain (Order Theory)",
"Definition:Upper Bound of Set",
"Definition:Non-Empty Set",
"Definition:Chain (Order Theory)",
"Definition:Chain (Order Theory)",
"Definition:Linearly Independent/Set",
"Definition:Linearly Independent/Set",
"Definitio... |
proofwiki-20419 | Linearly Independent Set is Contained in some Basis/Finite Dimensional Case | Let $E$ be a vector space of $n$ dimensions.
Let $H$ be a linearly independent subset of $E$.
There exists a basis $B$ for $E$ such that $H \subseteq B$. | By hypothesis there is a basis $B$ of $E$ with $n$ elements.
Then $H \cup B$ is a generator for $E$.
So by Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set there exists a basis $C$ of $E$ such that $H \subseteq C \subseteq H \cup B$.
{{Qed}} | Let $E$ be a [[Definition:Vector Space|vector space]] of $n$ [[Definition:Dimension of Vector Space|dimensions]].
Let $H$ be a [[Definition:Linearly Independent Set|linearly independent subset]] of $E$.
There exists a [[Definition:Basis of Vector Space|basis]] $B$ for $E$ such that $H \subseteq B$. | [[Definition:By Hypothesis|By hypothesis]] there is a [[Definition:Basis of Vector Space|basis]] $B$ of $E$ with $n$ [[Definition:Element|elements]].
Then $H \cup B$ is a [[Definition:Generator of Module|generator]] for $E$.
So by [[Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set]] the... | Linearly Independent Set is Contained in some Basis/Finite Dimensional Case/Proof 1 | https://proofwiki.org/wiki/Linearly_Independent_Set_is_Contained_in_some_Basis/Finite_Dimensional_Case | https://proofwiki.org/wiki/Linearly_Independent_Set_is_Contained_in_some_Basis/Finite_Dimensional_Case/Proof_1 | [
"Linearly Independent Set is Contained in some Basis"
] | [
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Linearly Independent/Set",
"Definition:Basis of Vector Space"
] | [
"Definition:By Hypothesis",
"Definition:Basis of Vector Space",
"Definition:Element",
"Definition:Generator of Module",
"Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set",
"Definition:Basis of Vector Space"
] |
proofwiki-20420 | Linearly Independent Set is Contained in some Basis/Finite Dimensional Case | Let $E$ be a vector space of $n$ dimensions.
Let $H$ be a linearly independent subset of $E$.
There exists a basis $B$ for $E$ such that $H \subseteq B$. | Let $H = \set {\xi_1, \xi_2, \ldots, \xi_r}$.
Consider the basis $B = \set {\alpha_1, \alpha_2, \ldots, \alpha_n}$ of $E$.
Consider the set $G = H \cup B = \set {\xi_1, \xi_2, \ldots, \xi_r, \alpha_1, \alpha_2, \ldots, \alpha_n}$.
We have that $G$ is a generator of $E$.
As $B$ is a basis, it follows that each of $H$ is... | Let $E$ be a [[Definition:Vector Space|vector space]] of $n$ [[Definition:Dimension of Vector Space|dimensions]].
Let $H$ be a [[Definition:Linearly Independent Set|linearly independent subset]] of $E$.
There exists a [[Definition:Basis of Vector Space|basis]] $B$ for $E$ such that $H \subseteq B$. | Let $H = \set {\xi_1, \xi_2, \ldots, \xi_r}$.
Consider the [[Definition:Basis of Vector Space|basis]] $B = \set {\alpha_1, \alpha_2, \ldots, \alpha_n}$ of $E$.
Consider the [[Definition:Set|set]] $G = H \cup B = \set {\xi_1, \xi_2, \ldots, \xi_r, \alpha_1, \alpha_2, \ldots, \alpha_n}$.
We have that $G$ is a [[Defini... | Linearly Independent Set is Contained in some Basis/Finite Dimensional Case/Proof 2 | https://proofwiki.org/wiki/Linearly_Independent_Set_is_Contained_in_some_Basis/Finite_Dimensional_Case | https://proofwiki.org/wiki/Linearly_Independent_Set_is_Contained_in_some_Basis/Finite_Dimensional_Case/Proof_2 | [
"Linearly Independent Set is Contained in some Basis"
] | [
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Linearly Independent/Set",
"Definition:Basis of Vector Space"
] | [
"Definition:Basis of Vector Space",
"Definition:Set",
"Definition:Generator of Vector Space",
"Definition:Basis of Vector Space",
"Definition:Linear Combination",
"Definition:Linearly Dependent/Set",
"Definition:Element",
"Definition:Linear Combination",
"Definition:Element",
"Definition:Set",
"... |
proofwiki-20421 | Generator of Vector Space Contains Basis/Infinite Dimensional Case | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $G$ be a generator of $X$.
Then:
:$G$ contains a basis for $X$. | If $X = \set { {\mathbf 0}_X}$, then we must have $G \subseteq \set { {\mathbf 0}_X}$, while the only basis for $X$ is $\O$.
So in this case, we have the claim immediately.
Now take $X \ne \set { {\mathbf 0}_X}$.
Let:
:$\SS = \set {L \subseteq G : L \text { is linearly independent} }$
Since $G$ generates $X$, it con... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $G$ be a [[Definition:Generator of Module|generator]] of $X$.
Then:
:$G$ contains a [[Definition:Basis of Vector Space|basis]] for $X$. | If $X = \set { {\mathbf 0}_X}$, then we must have $G \subseteq \set { {\mathbf 0}_X}$, while the only [[Definition:Basis of Vector Space|basis]] for $X$ is $\O$.
So in this case, we have the claim immediately.
Now take $X \ne \set { {\mathbf 0}_X}$.
Let:
:$\SS = \set {L \subseteq G : L \text { is linearly indepe... | Generator of Vector Space Contains Basis/Infinite Dimensional Case | https://proofwiki.org/wiki/Generator_of_Vector_Space_Contains_Basis/Infinite_Dimensional_Case | https://proofwiki.org/wiki/Generator_of_Vector_Space_Contains_Basis/Infinite_Dimensional_Case | [
"Generator of Vector Space Contains Basis"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Generator of Module",
"Definition:Basis of Vector Space"
] | [
"Definition:Basis of Vector Space",
"Definition:Generator of Module",
"Zorn's Lemma",
"Definition:Non-Empty Set",
"Definition:Chain (Order Theory)",
"Definition:Upper Bound of Set",
"Definition:Non-Empty Set",
"Definition:Chain (Order Theory)",
"Definition:Chain (Order Theory)",
"Definition:Linear... |
proofwiki-20422 | Vector not contained in Linear Span of Linearly Independent Set is Linearly Independent of Set | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $L$ be a linearly independent set of $X$ such that:
:$U = \map \span L \ne X$
Let:
:$x \in X \setminus U$
Then $L \cup \set x$ is linearly independent. | Take $x_1, \ldots, x_n \in L$ and take $\alpha_1, \ldots, \alpha_n, \alpha_{n + 1} \in K$ such that:
:$\ds \alpha_{n + 1} x + \sum_{k \mathop = 1}^n \alpha_i x_i = 0$
If $\alpha_{n + 1} = 0$, then we have:
:$\ds \sum_{k \mathop = 1}^n \alpha_i x_i = 0$
and so $\alpha_1 = \alpha_2 = \ldots = \alpha_n = 0$ from the lin... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $L$ be a [[Definition:Linearly Independent Set|linearly independent set]] of $X$ such that:
:$U = \map \span L \ne X$
Let:
:$x \in X \setminus U$
Then $L \cup \set x$ is [[Definition... | Take $x_1, \ldots, x_n \in L$ and take $\alpha_1, \ldots, \alpha_n, \alpha_{n + 1} \in K$ such that:
:$\ds \alpha_{n + 1} x + \sum_{k \mathop = 1}^n \alpha_i x_i = 0$
If $\alpha_{n + 1} = 0$, then we have:
:$\ds \sum_{k \mathop = 1}^n \alpha_i x_i = 0$
and so $\alpha_1 = \alpha_2 = \ldots = \alpha_n = 0$ from the... | Vector not contained in Linear Span of Linearly Independent Set is Linearly Independent of Set | https://proofwiki.org/wiki/Vector_not_contained_in_Linear_Span_of_Linearly_Independent_Set_is_Linearly_Independent_of_Set | https://proofwiki.org/wiki/Vector_not_contained_in_Linear_Span_of_Linearly_Independent_Set_is_Linearly_Independent_of_Set | [
"Linear Independence"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Linearly Independent/Set",
"Definition:Linearly Independent/Set"
] | [
"Definition:Linearly Independent/Set",
"Linear Span is Linear Subspace",
"Definition:Linearly Independent/Set",
"Category:Linear Independence"
] |
proofwiki-20423 | Dimension of Image of Vector Space under Linear Transformation is Bounded Above by Dimension of Vector Space | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $T : X \to Y$ be a linear transformation.
Then:
:$\dim T \sqbrk X \le \dim X$ | From Vector Space has Basis, there exists a basis $\BB$ for $X$.
By Image of Generating Set of Vector Space under Linear Transformation is Generating Set of Image, $T \sqbrk \BB$ is a generator for $T \sqbrk X$.
From Generator of Vector Space Contains Basis, there exists a basis $\BB'$ for $T \sqbrk X$ such that $\BB' ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Then:
:$\dim T \sqbrk X \le \dim X$ | From [[Vector Space has Basis]], there exists a [[Definition:Basis of Vector Space|basis]] $\BB$ for $X$.
By [[Image of Generating Set of Vector Space under Linear Transformation is Generating Set of Image]], $T \sqbrk \BB$ is a [[Definition:Generator of Module|generator]] for $T \sqbrk X$.
From [[Generator of Vector... | Dimension of Image of Vector Space under Linear Transformation is Bounded Above by Dimension of Vector Space | https://proofwiki.org/wiki/Dimension_of_Image_of_Vector_Space_under_Linear_Transformation_is_Bounded_Above_by_Dimension_of_Vector_Space | https://proofwiki.org/wiki/Dimension_of_Image_of_Vector_Space_under_Linear_Transformation_is_Bounded_Above_by_Dimension_of_Vector_Space | [
"Dimension of Image of Vector Space under Linear Transformation is Bounded Above by Dimension of Vector Space",
"Linear Transformations"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Linear Transformation"
] | [
"Vector Space has Basis",
"Definition:Basis of Vector Space",
"Image of Generating Set of Vector Space under Linear Transformation is Generating Set of Image",
"Definition:Generator of Module",
"Generator of Vector Space Contains Basis",
"Definition:Basis of Vector Space",
"Cardinality of Image of Mappi... |
proofwiki-20424 | Image of Generating Set of Vector Space under Linear Transformation is Generating Set of Image | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $T : X \to Y$ be a linear transformation.
Let $G$ be a generating set for $X$.
Then $T \sqbrk G$ be a generating set for $T \sqbrk X$. | Let $y \in T \sqbrk X$.
Then there exists $x \in X$ such that $y = T x$.
Since $G$ generates $X$, there exists $n \in \N$, $x_1, \ldots, x_n \in G$ and $\alpha_1, \ldots, \alpha_n \in K$ such that:
:$\ds x = \sum_{i \mathop = 1}^n \alpha_i x_i$
Then from the linearity of $T$ we have:
:$\ds y = T x = \sum_{i \mathop ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Let $G$ be a [[Definition:Generator of Module|generating set]] for $X$.
Then $T \sqbrk G$ be a [[Definitio... | Let $y \in T \sqbrk X$.
Then there exists $x \in X$ such that $y = T x$.
Since $G$ [[Definition:Generator of Module|generates]] $X$, there exists $n \in \N$, $x_1, \ldots, x_n \in G$ and $\alpha_1, \ldots, \alpha_n \in K$ such that:
:$\ds x = \sum_{i \mathop = 1}^n \alpha_i x_i$
Then from the [[Definition:Linear ... | Image of Generating Set of Vector Space under Linear Transformation is Generating Set of Image | https://proofwiki.org/wiki/Image_of_Generating_Set_of_Vector_Space_under_Linear_Transformation_is_Generating_Set_of_Image | https://proofwiki.org/wiki/Image_of_Generating_Set_of_Vector_Space_under_Linear_Transformation_is_Generating_Set_of_Image | [
"Linear Transformations"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Linear Transformation",
"Definition:Generator of Module",
"Definition:Generator of Module"
] | [
"Definition:Generator of Module",
"Definition:Linear Transformation",
"Definition:Generator of Module",
"Category:Linear Transformations"
] |
proofwiki-20425 | Linear Functional on Vector Space is Zero or Surjective | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $f : X \to K$ be a linear functional.
Then either:
:$\map f x = 0$ for each $x \in X$
or:
:$f$ is surjective. | Suppose that $\map f {x_0} \ne 0$ for $x_0 \in X$.
Take $c \in K$.
Then we have, from linearity:
{{begin-eqn}}
{{eqn | l = \map f {c \paren {\map f {x_0} }^{-1} x_0}
| r = c \paren {\map f {x_0} }^{-1} \map f {x_0}
}}
{{eqn | r = c
}}
{{end-eqn}}
Since $c \in K$ was arbitrary, we have that $f$ is surjective.
{{qed}... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $f : X \to K$ be a [[Definition:Linear Functional|linear functional]].
Then either:
:$\map f x = 0$ for each $x \in X$
or:
:$f$ is [[Definition:Surjective Mapping|surjective]]. | Suppose that $\map f {x_0} \ne 0$ for $x_0 \in X$.
Take $c \in K$.
Then we have, from [[Definition:Linear Functional|linearity]]:
{{begin-eqn}}
{{eqn | l = \map f {c \paren {\map f {x_0} }^{-1} x_0}
| r = c \paren {\map f {x_0} }^{-1} \map f {x_0}
}}
{{eqn | r = c
}}
{{end-eqn}}
Since $c \in K$ was arbitrary, w... | Linear Functional on Vector Space is Zero or Surjective | https://proofwiki.org/wiki/Linear_Functional_on_Vector_Space_is_Zero_or_Surjective | https://proofwiki.org/wiki/Linear_Functional_on_Vector_Space_is_Zero_or_Surjective | [
"Linear Functionals"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Linear Functional",
"Definition:Surjection"
] | [
"Definition:Linear Functional",
"Definition:Surjection",
"Category:Linear Functionals"
] |
proofwiki-20426 | Continuity of Linear Functionals in Weak Topology Induced by Pair of Vector Spaces with Bilinear Mapping | Let $\GF \in \set {\R, \C}$.
Let $X$ and $X'$ be vector spaces over $\GF$.
Let $\innerprod \cdot \cdot : X \times X' \to \GF$ be a bilinear mapping.
For each $x' \in X'$, define $f_{x'} : X \to \GF$ by:
:$\map {f_{x'} } x = \innerprod x {x'}$ for each $x \in X$.
Let:
:$F = \set {f_{x'} : x' \in X'}$
Let $\map \sigma {... | === Sufficient Condition ===
From the definition of the weak topology, $\map \sigma {X, X'}$ is the coarsest topology making all $g \in F$ continuous.
So if $g \in F$ then $g$ is $\map \sigma {X, X'}$-continuous.
{{qed|lemma}} | Let $\GF \in \set {\R, \C}$.
Let $X$ and $X'$ be [[Definition:Vector Space|vector spaces]] over $\GF$.
Let $\innerprod \cdot \cdot : X \times X' \to \GF$ be a [[Definition:Bilinear Mapping|bilinear mapping]].
For each $x' \in X'$, define $f_{x'} : X \to \GF$ by:
:$\map {f_{x'} } x = \innerprod x {x'}$ for each $x \... | === Sufficient Condition ===
From the definition of the [[Definition:Weak Topology Induced by Dual System|weak topology]], $\map \sigma {X, X'}$ is the [[Definition:Coarser Topology|coarsest topology]] making all $g \in F$ [[Definition:Continuous Mapping (Topology)|continuous]].
So if $g \in F$ then $g$ is [[Definiti... | Continuity of Linear Functionals in Weak Topology Induced by Pair of Vector Spaces with Bilinear Mapping | https://proofwiki.org/wiki/Continuity_of_Linear_Functionals_in_Weak_Topology_Induced_by_Pair_of_Vector_Spaces_with_Bilinear_Mapping | https://proofwiki.org/wiki/Continuity_of_Linear_Functionals_in_Weak_Topology_Induced_by_Pair_of_Vector_Spaces_with_Bilinear_Mapping | [
"Initial Topology",
"Dual Systems"
] | [
"Definition:Vector Space",
"Definition:Bilinear Mapping",
"Definition:Initial Topology",
"Definition:Linear Functional",
"Definition:Continuous Mapping"
] | [
"Definition:Weak Topology Induced by Dual System",
"Definition:Coarser Topology",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)"
] |
proofwiki-20427 | Frink's Metrization Theorem | Let $T = \struct {S, \tau}$ be a topological space.
Then:
:$T$ is metrizable
{{iff}}:
:for all $s \in S$ there exists a countable neighborhood basis, denoted $\set {U_{s, n} : n \in \N}$, such that:
::$(1): \quad \forall s \in S, n \in \N : U_{s, n + 1} \subseteq U_{s, n}$
::$(2): \quad \forall s \in S, n \in \N : \exi... | {{proof wanted}}
{{Namedfor|Aline Huke Frink|cat = Frink}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then:
:$T$ is [[Definition:Metrizable Space|metrizable]]
{{iff}}:
:for all $s \in S$ there exists a [[Definition:Countable Set|countable]] [[Definition:Neighborhood Basis|neighborhood basis]], denoted $\set {U_{s, n} : n \in \N}$, su... | {{proof wanted}}
{{Namedfor|Aline Huke Frink|cat = Frink}} | Frink's Metrization Theorem | https://proofwiki.org/wiki/Frink's_Metrization_Theorem | https://proofwiki.org/wiki/Frink's_Metrization_Theorem | [
"Metrization Theorems",
"Neighborhood Bases",
"Metrizable Spaces"
] | [
"Definition:Topological Space",
"Definition:Metrizable Space",
"Definition:Countable Set",
"Definition:Neighborhood Basis"
] | [] |
proofwiki-20428 | Linear Functional on Complex Vector Space is Uniquely Determined by Real Part | Let $X$ be a vector space over $\C$.
Let $f : X \to \C$ be a linear functional.
Define a function $g : X \to \R$:
:$\map g x = \map \Re {\map f x}$
for each $x \in X$.
Then:
:$\map f x = \map g x - i \map g {i x}$
for each $x \in X$. | For brevity, define a function $h : X \to \R$ by:
:$\map h x = \map \Im {\map f x}$
for each $x \in X$.
Note that:
:$\map f x = \map \Re {\map f x} + i \map \Im {\map f x} = \map g x + i \map h x$
so that:
:$\map f {i x} = \map g {i x} + i \map h {i x}$
for each $x \in X$.
On the other hand, by the linearity of $f$,... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\C$.
Let $f : X \to \C$ be a [[Definition:Linear Functional|linear functional]].
Define a [[Definition:Function|function]] $g : X \to \R$:
:$\map g x = \map \Re {\map f x}$
for each $x \in X$.
Then:
:$\map f x = \map g x - i \map g {i x}$
for each $... | For brevity, define a [[Definition:Function|function]] $h : X \to \R$ by:
:$\map h x = \map \Im {\map f x}$
for each $x \in X$.
Note that:
:$\map f x = \map \Re {\map f x} + i \map \Im {\map f x} = \map g x + i \map h x$
so that:
:$\map f {i x} = \map g {i x} + i \map h {i x}$
for each $x \in X$.
On the othe... | Linear Functional on Complex Vector Space is Uniquely Determined by Real Part | https://proofwiki.org/wiki/Linear_Functional_on_Complex_Vector_Space_is_Uniquely_Determined_by_Real_Part | https://proofwiki.org/wiki/Linear_Functional_on_Complex_Vector_Space_is_Uniquely_Determined_by_Real_Part | [
"Linear Functionals"
] | [
"Definition:Vector Space",
"Definition:Linear Functional",
"Definition:Function"
] | [
"Definition:Function",
"Definition:Linear Functional",
"Definition:Complex Number/Real Part",
"Category:Linear Functionals"
] |
proofwiki-20429 | Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube/Lemma 1 | :$\AA$ is countable | We have:
:$\AA \subseteq \BB \times \BB$
where $\BB \times \BB$ is the Cartesian product of $\BB$ with itself.
From Cartesian Product of Countable Sets is Countable:
:$\BB \times \BB$ is countable
From Subset of Countable Set is Countable:
:$\AA$ is countable
{{qed}}
Category:Regular Second-Countable Space is Homeomorp... | :$\AA$ is [[Definition:Countable Set|countable]] | We have:
:$\AA \subseteq \BB \times \BB$
where $\BB \times \BB$ is the [[Definition:Cartesian Product|Cartesian product]] of $\BB$ with itself.
From [[Cartesian Product of Countable Sets is Countable]]:
:$\BB \times \BB$ is [[Definition:Countable Set|countable]]
From [[Subset of Countable Set is Countable]]:
:$\AA$ i... | Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube/Lemma 1 | https://proofwiki.org/wiki/Regular_Second-Countable_Space_is_Homeomorphic_to_Subspace_of_Hilbert_Cube/Lemma_1 | https://proofwiki.org/wiki/Regular_Second-Countable_Space_is_Homeomorphic_to_Subspace_of_Hilbert_Cube/Lemma_1 | [
"Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube"
] | [
"Definition:Countable Set"
] | [
"Definition:Cartesian Product",
"Cartesian Product of Countable Sets is Countable",
"Definition:Countable Set",
"Subset of Countable Set is Countable",
"Definition:Countable Set",
"Category:Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube"
] |
proofwiki-20430 | Regular Lindelöf Space is Normal | Let $T = \struct {S, \tau}$ be a regular Lindelöf topological space.
Then $T$ is a normal space. | {{Recall|Normal Space|normal space|index = 1}}
{{Definition:Normal Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a regular Lindelöf space.
{{Recall|Regular Space|regular space|index = 2}}
{{Definition:Regular Space/Definition 2}}
From $T_3$ Lindelöf Space is $T_4$:
:$T$ is a $T_4$ space
By definition, $T$ is a no... | Let $T = \struct {S, \tau}$ be a [[Definition:Regular Space|regular]] [[Definition:Lindelöf Space|Lindelöf]] [[Definition:Topological Space|topological space]].
Then $T$ is a [[Definition:Normal Space|normal space]]. | {{Recall|Normal Space|normal space|index = 1}}
{{Definition:Normal Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a [[Definition:Regular Space|regular]] [[Definition:Lindelöf Space|Lindelöf space]].
{{Recall|Regular Space|regular space|index = 2}}
{{Definition:Regular Space/Definition 2}}
From [[T3 Lindelöf Spa... | Regular Lindelöf Space is Normal | https://proofwiki.org/wiki/Regular_Lindelöf_Space_is_Normal | https://proofwiki.org/wiki/Regular_Lindelöf_Space_is_Normal | [
"Regular Spaces",
"Lindelöf Spaces",
"Normal Spaces"
] | [
"Definition:Regular Space",
"Definition:Lindelöf Space",
"Definition:Topological Space",
"Definition:Normal Space"
] | [
"Definition:Regular Space",
"Definition:Lindelöf Space",
"T3 Lindelöf Space is T4",
"Definition:T4 Space",
"Definition:Normal Space",
"Category:Regular Spaces",
"Category:Lindelöf Spaces",
"Category:Normal Spaces"
] |
proofwiki-20431 | Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube | Let $T = \struct {S, \tau}$ be a topological space which is regular and second-countable.
Then $T$ is homeomorphic to a subspace of the Hilbert cube. | From Second-Countable Space is Lindelöf:
:$T$ is a Lindelöf space
From Regular Lindelöf Space is Normal:
:$T$ is a normal space
By definition of second-countable:
:there exists a countable basis $\BB$ for $\tau$
Let:
:$\AA = \set {\tuple {U, V} : U, V \in \BB : U^- \subseteq V}$
where $U^-$ denotes the closure of $U$ ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Regular Space|regular]] and [[Definition:Second-Countable Space|second-countable]].
Then $T$ is [[Definition:Homeomorphism|homeomorphic]] to a [[Definition:Topological Subspace|subspace]] of the [[Definition:Hilbe... | From [[Second-Countable Space is Lindelöf]]:
:$T$ is a [[Definition:Lindelöf Space|Lindelöf space]]
From [[Regular Lindelöf Space is Normal]]:
:$T$ is a [[Definition:Normal Space|normal space]]
By definition of [[Definition:Second-Countable Space|second-countable]]:
:there exists a [[Definition:Countable Set|counta... | Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube | https://proofwiki.org/wiki/Regular_Second-Countable_Space_is_Homeomorphic_to_Subspace_of_Hilbert_Cube | https://proofwiki.org/wiki/Regular_Second-Countable_Space_is_Homeomorphic_to_Subspace_of_Hilbert_Cube | [
"Regular Spaces",
"Second-Countable Spaces",
"Hilbert Cube",
"Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube"
] | [
"Definition:Topological Space",
"Definition:Regular Space",
"Definition:Second-Countable Space",
"Definition:Homeomorphism",
"Definition:Topological Subspace",
"Definition:Hilbert Cube"
] | [
"Second-Countable Space is Lindelöf",
"Definition:Lindelöf Space",
"Regular Lindelöf Space is Normal",
"Definition:Normal Space",
"Definition:Second-Countable Space",
"Definition:Countable Set",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Closure (Topology)",
"Definition:Countable Set"... |
proofwiki-20432 | Equivalence Classes of Diagonal Relation | Let $S$ be a set.
Let $\Delta_S$ denote the diagonal relation on $S$.
The set $\EE_S$ of equivalence classes of $S$ can be expressed as:
:$\EE_S = \set {\set x: x \in S}$
That is, it is the set of all singletons of $S$. | Let $x \in S$.
Then by definition of the diagonal relation:
:$y \mathrel {\Delta_S} x \iff y = x$
Hence:
:$y \in \eqclass x {\Delta_S} \iff y = x$
That is:
:$\eqclass x {\Delta_S} = \set x$
Hence the result.
{{qed}}
Category:Diagonal Relation
Category:Examples of Equivalence Classes
9hc4lfljm0a9bjr4xhjdjcp465t6x07 | Let $S$ be a [[Definition:Set|set]].
Let $\Delta_S$ denote the [[Definition:Diagonal Relation|diagonal relation]] on $S$.
The [[Definition:Set|set]] $\EE_S$ of [[Definition:Equivalence Class|equivalence classes]] of $S$ can be expressed as:
:$\EE_S = \set {\set x: x \in S}$
That is, it is the [[Definition:Set|set]]... | Let $x \in S$.
Then by definition of the [[Definition:Diagonal Relation|diagonal relation]]:
:$y \mathrel {\Delta_S} x \iff y = x$
Hence:
:$y \in \eqclass x {\Delta_S} \iff y = x$
That is:
:$\eqclass x {\Delta_S} = \set x$
Hence the result.
{{qed}}
[[Category:Diagonal Relation]]
[[Category:Examples of Equivalence ... | Equivalence Classes of Diagonal Relation | https://proofwiki.org/wiki/Equivalence_Classes_of_Diagonal_Relation | https://proofwiki.org/wiki/Equivalence_Classes_of_Diagonal_Relation | [
"Diagonal Relation",
"Examples of Equivalence Classes"
] | [
"Definition:Set",
"Definition:Diagonal Relation",
"Definition:Set",
"Definition:Equivalence Class",
"Definition:Set",
"Definition:Singleton"
] | [
"Definition:Diagonal Relation",
"Category:Diagonal Relation",
"Category:Examples of Equivalence Classes"
] |
proofwiki-20433 | Set of Ordered Pairs of Integers is Countable Infinite | The set of all ordered pairs of integers $\Z$ is countably infinite. | The set of all ordered pairs of a set $S$ is by definition the Cartesian product $S \times S$.
In this context we are determining the cardinality of $\Z \times \Z$.
From Integers are Countably Infinite, we have that $\Z$ is a countably infinite set.
The result then follows from Cartesian Product of Countable Sets is Co... | The [[Definition:Set|set]] of all [[Definition:Ordered Pair|ordered pairs]] of [[Definition:Integer|integers]] $\Z$ is [[Definition:Countably Infinite Set|countably infinite]]. | The [[Definition:Set|set]] of all [[Definition:Ordered Pair|ordered pairs]] of a [[Definition:Set|set]] $S$ is by definition the [[Definition:Cartesian Product|Cartesian product]] $S \times S$.
In this context we are determining the [[Definition:Cardinality|cardinality]] of $\Z \times \Z$.
From [[Integers are Countab... | Set of Ordered Pairs of Integers is Countable Infinite | https://proofwiki.org/wiki/Set_of_Ordered_Pairs_of_Integers_is_Countable_Infinite | https://proofwiki.org/wiki/Set_of_Ordered_Pairs_of_Integers_is_Countable_Infinite | [
"Countable Sets",
"Integers"
] | [
"Definition:Set",
"Definition:Ordered Pair",
"Definition:Integer",
"Definition:Countably Infinite/Set"
] | [
"Definition:Set",
"Definition:Ordered Pair",
"Definition:Set",
"Definition:Cartesian Product",
"Definition:Cardinality",
"Integers are Countably Infinite",
"Definition:Countably Infinite/Set",
"Cartesian Product of Countable Sets is Countable"
] |
proofwiki-20434 | Union of Relation with Inverse is Symmetric Relation | Let $\RR$ be a relation on a set $S$.
Let $\RR^{-1}$ denote the inverse of $\RR$.
Then $\RR \cup \RR^{-1}$, the union of $\RR$ with $\RR^{-1}$, is symmetric. | Let $\tuple {a, b} \in \RR \cup \RR^{-1}$.
By definition of union, either:
:$\tuple {a, b} \in \RR$
or:
:$\tuple {a, b} \in \RR^{-1}$
;Case 1
If $\tuple {a, b} \in \RR$, then by definition of inverse relation:
:$\tuple {b, a} \in \RR^{-1}$
But from Set is Subset of Union:
:$\tuple {b, a} \in \RR \cup \RR^{-1}$
{{qed|le... | Let $\RR$ be a [[Definition:Relation|relation]] on a [[Definition:Set|set]] $S$.
Let $\RR^{-1}$ denote the [[Definition:Inverse Relation|inverse]] of $\RR$.
Then $\RR \cup \RR^{-1}$, the [[Definition:Set Union|union]] of $\RR$ with $\RR^{-1}$, is [[Definition:Symmetric Relation|symmetric]]. | Let $\tuple {a, b} \in \RR \cup \RR^{-1}$.
By definition of [[Definition:Set Union|union]], either:
:$\tuple {a, b} \in \RR$
or:
:$\tuple {a, b} \in \RR^{-1}$
;Case 1
If $\tuple {a, b} \in \RR$, then by definition of [[Definition:Inverse Relation|inverse relation]]:
:$\tuple {b, a} \in \RR^{-1}$
But from [[Set ... | Union of Relation with Inverse is Symmetric Relation | https://proofwiki.org/wiki/Union_of_Relation_with_Inverse_is_Symmetric_Relation | https://proofwiki.org/wiki/Union_of_Relation_with_Inverse_is_Symmetric_Relation | [
"Set Union",
"Inverse Relations",
"Symmetric Relations"
] | [
"Definition:Relation",
"Definition:Set",
"Definition:Inverse Relation",
"Definition:Set Union",
"Definition:Symmetric Relation"
] | [
"Definition:Set Union",
"Definition:Inverse Relation",
"Set is Subset of Union",
"Definition:Inverse Relation",
"Inverse of Inverse Relation",
"Set is Subset of Union",
"Proof by Cases",
"Definition:Symmetric Relation",
"Category:Set Union",
"Category:Inverse Relations",
"Category:Symmetric Rela... |
proofwiki-20435 | Equivalence of Definitions of Symmetric Closure | {{TFAE|def = Symmetric Closure}}
Let $\RR$ be a relation on a set $S$. | First we note that from Union of Relation with Inverse is Symmetric Relation, $\RR \cup \RR^{-1}$ is a symmetric relation. | {{TFAE|def = Symmetric Closure}}
Let $\RR$ be a [[Definition:Relation|relation]] on a [[Definition:Set|set]] $S$. | First we note that from [[Union of Relation with Inverse is Symmetric Relation]], $\RR \cup \RR^{-1}$ is a [[Definition:Symmetric Relation|symmetric relation]]. | Equivalence of Definitions of Symmetric Closure | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Closure | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Closure | [
"Symmetric Closures"
] | [
"Definition:Relation",
"Definition:Set"
] | [
"Union of Relation with Inverse is Symmetric Relation",
"Definition:Symmetric Relation",
"Definition:Symmetric Relation",
"Definition:Symmetric Relation",
"Definition:Symmetric Relation",
"Definition:Symmetric Relation",
"Definition:Symmetric Relation",
"Definition:Symmetric Relation"
] |
proofwiki-20436 | Real Part of Linear Functional is Linear Functional | Let $X$ be a vector space over $\C$.
Let $f : X \to \C$ be a linear functional.
Define $g : X \to \R$ by:
:$\map g x = \map \Re {\map f x}$
for each $x \in X$.
Then $f$ is $\R$-linear. | Let $x, y \in X$ and $\lambda, \mu \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \map g {\lambda x + \mu y}
| r = \map \Re {\map f {\lambda x + \mu y} }
}}
{{eqn | r = \frac 1 2 \paren {\map f {\lambda x + \mu y} + \overline {\map f {\lambda x + \mu y} } }
| c = Sum of Complex Number with Conjugate
}}
{{eqn | r = \frac... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\C$.
Let $f : X \to \C$ be a [[Definition:Linear Functional|linear functional]].
Define $g : X \to \R$ by:
:$\map g x = \map \Re {\map f x}$
for each $x \in X$.
Then $f$ is [[Definition:Linear Functional|$\R$-linear]]. | Let $x, y \in X$ and $\lambda, \mu \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \map g {\lambda x + \mu y}
| r = \map \Re {\map f {\lambda x + \mu y} }
}}
{{eqn | r = \frac 1 2 \paren {\map f {\lambda x + \mu y} + \overline {\map f {\lambda x + \mu y} } }
| c = [[Sum of Complex Number with Conjugate]]
}}
{{eqn | r =... | Real Part of Linear Functional is Linear Functional | https://proofwiki.org/wiki/Real_Part_of_Linear_Functional_is_Linear_Functional | https://proofwiki.org/wiki/Real_Part_of_Linear_Functional_is_Linear_Functional | [
"Linear Functionals"
] | [
"Definition:Vector Space",
"Definition:Linear Functional",
"Definition:Linear Functional"
] | [
"Sum of Complex Number with Conjugate",
"Sum of Complex Conjugates",
"Definition:Linear Functional",
"Product of Complex Conjugates",
"Category:Linear Functionals"
] |
proofwiki-20437 | Imaginary Part of Linear Functional is Linear Functional | Let $X$ be a vector space over $\C$.
Let $f : X \to \C$ be a linear functional.
Define $h : X \to \R$ by:
:$\map h x = \map \Im {\map f x}$
for each $x \in X$.
Then $f$ is $\R$-linear. | Let $x, y \in X$ and $\lambda, \mu \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \map h {\lambda x + \mu y}
| r = \map \Re {\map f {\lambda x + \mu y} }
}}
{{eqn | r = \frac 1 {2 i} \paren {\map f {\lambda x + \mu y} - \overline {\map f {\lambda x + \mu y} } }
| c = Difference of Complex Number with Conjugate
}}
{{eqn ... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\C$.
Let $f : X \to \C$ be a [[Definition:Linear Functional|linear functional]].
Define $h : X \to \R$ by:
:$\map h x = \map \Im {\map f x}$
for each $x \in X$.
Then $f$ is [[Definition:Linear Functional|$\R$-linear]]. | Let $x, y \in X$ and $\lambda, \mu \in \R$.
Then:
{{begin-eqn}}
{{eqn | l = \map h {\lambda x + \mu y}
| r = \map \Re {\map f {\lambda x + \mu y} }
}}
{{eqn | r = \frac 1 {2 i} \paren {\map f {\lambda x + \mu y} - \overline {\map f {\lambda x + \mu y} } }
| c = [[Difference of Complex Number with Conjugate]]
}}
... | Imaginary Part of Linear Functional is Linear Functional | https://proofwiki.org/wiki/Imaginary_Part_of_Linear_Functional_is_Linear_Functional | https://proofwiki.org/wiki/Imaginary_Part_of_Linear_Functional_is_Linear_Functional | [
"Linear Functionals"
] | [
"Definition:Vector Space",
"Definition:Linear Functional",
"Definition:Linear Functional"
] | [
"Difference of Complex Number with Conjugate",
"Sum of Complex Conjugates",
"Definition:Linear Functional",
"Product of Complex Conjugates",
"Category:Linear Functionals"
] |
proofwiki-20438 | Hahn-Banach Theorem for Continuous Linear Functional on Locally Convex Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a locally convex space over $\GF$ with its standard topology.
Let $X_0$ be a linear subspace of $X$.
Let $f_0 : X_0 \to \GF$ be a continuous linear functional.
Then there exists a continuous linear functional $f : X \to \GF$ such that $f$ extends $f_0$. | By Normed Vector Space is Locally Convex Space and Norm on Vector Space is Seminorm, we can view the normed vector space $\struct {\GF, \cmod {\, \cdot \,} }$ as the locally convex space $\struct {\GF, \set {\cmod {\, \cdot \,} } }$.
From Characterization of Continuous Linear Transformations between Locally Convex Spac... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Locally Convex Space|locally convex space]] over $\GF$ with its [[Definition:Locally Convex Space/Standard Topology|standard topology]].
Let $X_0$ be a [[Definition:Linear Subspace|linear subspace]] of $X$.
Let $f_0 : X_0 \to \GF$ be a [[Definit... | By [[Normed Vector Space is Locally Convex Space]] and [[Norm on Vector Space is Seminorm]], we can view the [[Definition:Normed Vector Space|normed vector space]] $\struct {\GF, \cmod {\, \cdot \,} }$ as the [[Definition:Locally Convex Space|locally convex space]] $\struct {\GF, \set {\cmod {\, \cdot \,} } }$.
From [... | Hahn-Banach Theorem for Continuous Linear Functional on Locally Convex Space | https://proofwiki.org/wiki/Hahn-Banach_Theorem_for_Continuous_Linear_Functional_on_Locally_Convex_Space | https://proofwiki.org/wiki/Hahn-Banach_Theorem_for_Continuous_Linear_Functional_on_Locally_Convex_Space | [
"Functional Analysis",
"Locally Convex Spaces"
] | [
"Definition:Locally Convex Space",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Linear Subspace",
"Definition:Continuous Mapping (Topology)",
"Definition:Linear Functional",
"Definition:Continuous Mapping (Topology)",
"Definition:Linear Functional",
"Definition:Extension of Mapping... | [
"Normed Vector Space is Locally Convex Space",
"Norm on Vector Space is Seminorm",
"Definition:Normed Vector Space",
"Definition:Locally Convex Space",
"Characterization of Continuous Linear Transformations between Locally Convex Spaces",
"Pointwise Maximum of Finite Family of Seminorms is Seminorm",
"N... |
proofwiki-20439 | Non-Negative Scalar Multiple of Seminorm on Vector Space is Seminorm | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $p$ be a seminorm on $X$.
Let $\alpha \in \R_{\ge 0}$.
Let $q = \alpha p$.
Then $q$ is a seminorm on $X$. | === {{SeminormAxiom|2}} ===
Let $x \in X$ and $k \in \GF$.
We have:
{{begin-eqn}}
{{eqn | l = \map q {k x}
| r = \alpha \map p {k x}
}}
{{eqn | r = \alpha \cmod k \map p x
| c = {{SeminormAxiom|2}} for $p$
}}
{{eqn | r = \cmod k \map q x
}}
{{end-eqn}}
{{qed|lemma}} | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $p$ be a [[Definition:Seminorm|seminorm]] on $X$.
Let $\alpha \in \R_{\ge 0}$.
Let $q = \alpha p$.
Then $q$ is a [[Definition:Seminorm|seminorm]] on $X$. | === {{SeminormAxiom|2}} ===
Let $x \in X$ and $k \in \GF$.
We have:
{{begin-eqn}}
{{eqn | l = \map q {k x}
| r = \alpha \map p {k x}
}}
{{eqn | r = \alpha \cmod k \map p x
| c = {{SeminormAxiom|2}} for $p$
}}
{{eqn | r = \cmod k \map q x
}}
{{end-eqn}}
{{qed|lemma}} | Non-Negative Scalar Multiple of Seminorm on Vector Space is Seminorm | https://proofwiki.org/wiki/Non-Negative_Scalar_Multiple_of_Seminorm_on_Vector_Space_is_Seminorm | https://proofwiki.org/wiki/Non-Negative_Scalar_Multiple_of_Seminorm_on_Vector_Space_is_Seminorm | [
"Seminorms"
] | [
"Definition:Vector Space",
"Definition:Seminorm",
"Definition:Seminorm"
] | [] |
proofwiki-20440 | Norm on Vector Space is Seminorm | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $\norm {\, \cdot \,}$ be a norm on $X$.
Then $\norm {\, \cdot \,}$ is a seminorm. | Note that {{NormAxiomVector|2}} and {{NormAxiomVector|3}} are precisely {{SeminormAxiom|2}} and {{SeminormAxiom|3}}.
{{qed}}
Category:Seminorms
Category:Normed Spaces
ituqwziuvtw1f7cdjuiayolrsbculmt | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $\norm {\, \cdot \,}$ be a [[Definition:Norm on Vector Space|norm]] on $X$.
Then $\norm {\, \cdot \,}$ is a [[Definition:Seminorm|seminorm]]. | Note that {{NormAxiomVector|2}} and {{NormAxiomVector|3}} are precisely {{SeminormAxiom|2}} and {{SeminormAxiom|3}}.
{{qed}}
[[Category:Seminorms]]
[[Category:Normed Spaces]]
ituqwziuvtw1f7cdjuiayolrsbculmt | Norm on Vector Space is Seminorm | https://proofwiki.org/wiki/Norm_on_Vector_Space_is_Seminorm | https://proofwiki.org/wiki/Norm_on_Vector_Space_is_Seminorm | [
"Seminorms",
"Seminorms",
"Normed Spaces"
] | [
"Definition:Vector Space",
"Definition:Norm/Vector Space",
"Definition:Seminorm"
] | [
"Category:Seminorms",
"Category:Normed Spaces"
] |
proofwiki-20441 | Positive Scalar Multiple of Norm on Vector Space is Norm | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.
Let $\alpha > 0$ be a real number.
Define $\norm {\, \cdot \,}' : X \to \R_{\ge 0}$ by:
:$\norm x' = \alpha \norm x$
for each $x \in X$.
Then $\norm {\, \cdot \,}'$ is a norm on $X$. | === {{NormAxiomVector|1|nolink}} ===
Suppose that $x \in X$ is such that:
:$\norm x' = 0$
Then we have:
:$\alpha \norm x = 0$
Since $\alpha > 0$, it follows that:
:$\norm x = 0$
From {{NormAxiomVector|1}}, we have $x = {\mathbf 0}_X$.
{{qed|lemma}} | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\alpha > 0$ be a [[Definition:Real Number|real number]].
Define $\norm {\, \cdot \,}' : X \to \R_{\ge 0}$ by:
:$\norm x' = \alpha \norm x$
for each $x \in X$.
Then... | === {{NormAxiomVector|1|nolink}} ===
Suppose that $x \in X$ is such that:
:$\norm x' = 0$
Then we have:
:$\alpha \norm x = 0$
Since $\alpha > 0$, it follows that:
:$\norm x = 0$
From {{NormAxiomVector|1}}, we have $x = {\mathbf 0}_X$.
{{qed|lemma}} | Positive Scalar Multiple of Norm on Vector Space is Norm | https://proofwiki.org/wiki/Positive_Scalar_Multiple_of_Norm_on_Vector_Space_is_Norm | https://proofwiki.org/wiki/Positive_Scalar_Multiple_of_Norm_on_Vector_Space_is_Norm | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Real Number",
"Definition:Norm/Vector Space"
] | [] |
proofwiki-20442 | Vector Addition on Locally Convex Space is Continuous | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a locally convex space over $\GF$ equipped with the standard topolgoy.
Let $\struct {X \times X, \tau}$ be the Cartesian product $X \times X$ equipped with the product topology.
Let $+ : \struct {X \times X, \tau} \to \struct {X, \PP}$ be the vector addition defin... | From the definition of the standard topolgoy, the topology on $\struct {X, \PP}$ has sub-basis:
:$\SS = \set {\map {B_p} {x, \epsilon} : p \in \PP, \, \epsilon > 0}$
where we define:
:$\map {B_p} {x, \epsilon} = \set {y \in X : \map p {y - x} < \epsilon}$
for each $p \in \PP$ and $\epsilon > 0$.
From Continuity Test ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Locally Convex Space|locally convex space]] over $\GF$ equipped with the [[Definition:Locally Convex Space/Standard Topology|standard topolgoy]].
Let $\struct {X \times X, \tau}$ be the [[Definition:Cartesian Product|Cartesian product]] $X \times ... | From the definition of the [[Definition:Locally Convex Space/Standard Topology|standard topolgoy]], the [[Definition:Topology|topology]] on $\struct {X, \PP}$ has [[Definition:Sub-Basis|sub-basis]]:
:$\SS = \set {\map {B_p} {x, \epsilon} : p \in \PP, \, \epsilon > 0}$
where we define:
:$\map {B_p} {x, \epsilon} = ... | Vector Addition on Locally Convex Space is Continuous | https://proofwiki.org/wiki/Vector_Addition_on_Locally_Convex_Space_is_Continuous | https://proofwiki.org/wiki/Vector_Addition_on_Locally_Convex_Space_is_Continuous | [
"Locally Convex Spaces",
"Vector Addition"
] | [
"Definition:Locally Convex Space",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Cartesian Product",
"Definition:Product Topology",
"Definition:Vector Addition/Vector Space",
"Definition:Continuous Mapping"
] | [
"Definition:Locally Convex Space/Standard Topology",
"Definition:Topology",
"Definition:Sub-Basis",
"Continuity Test using Sub-Basis",
"Union of Subsets is Subset",
"Natural Basis of Product Topology/Finite Product",
"Definition:Basis (Topology)",
"Definition:Topology",
"Definition:Set Union",
"Ca... |
proofwiki-20443 | Scalar Multiplication on Locally Convex Space is Continuous | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a locally convex space over $\GF$ equipped with the standard topolgoy.
Let $\struct {\GF \times X, \tau}$ be the Cartesian product $\GF \times X$ equipped with the product topology.
Let $\circ : \struct {\GF \times X, \tau} \to \struct {X, \PP}$ be the scalar mult... | From the definition of the standard topology, the topology on $\struct {X, \PP}$ has sub-basis:
:$\SS = \set {\map {B_p} {x, \epsilon} : p \in \PP, \, \epsilon > 0}$
where we define:
:$\map {B_p} {x, \epsilon} = \set {y \in X : \map p {y - x} < \epsilon}$
for each $p \in \PP$ and $\epsilon > 0$.
From Continuity Test ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Locally Convex Space|locally convex space]] over $\GF$ equipped with the [[Definition:Locally Convex Space/Standard Topology|standard topolgoy]].
Let $\struct {\GF \times X, \tau}$ be the [[Definition:Cartesian Product|Cartesian product]] $\GF \ti... | From the definition of the [[Definition:Locally Convex Space/Standard Topology|standard topology]], the [[Definition:Topology|topology]] on $\struct {X, \PP}$ has [[Definition:Sub-Basis|sub-basis]]:
:$\SS = \set {\map {B_p} {x, \epsilon} : p \in \PP, \, \epsilon > 0}$
where we define:
:$\map {B_p} {x, \epsilon} = ... | Scalar Multiplication on Locally Convex Space is Continuous | https://proofwiki.org/wiki/Scalar_Multiplication_on_Locally_Convex_Space_is_Continuous | https://proofwiki.org/wiki/Scalar_Multiplication_on_Locally_Convex_Space_is_Continuous | [
"Locally Convex Spaces",
"Scalar Multiplication"
] | [
"Definition:Locally Convex Space",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Cartesian Product",
"Definition:Product Topology",
"Definition:Scalar Multiplication/Vector Space",
"Definition:Continuous Mapping"
] | [
"Definition:Locally Convex Space/Standard Topology",
"Definition:Topology",
"Definition:Sub-Basis",
"Continuity Test using Sub-Basis",
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Union of Subsets is Subset",
"Natural Basis of Product Topology/Finite Product... |
proofwiki-20444 | Locally Convex Space is Topological Vector Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a locally convex space over $\GF$ equipped with the standard topology $\tau$.
Then $\struct {X, \tau}$ is a topological vector space. | From Vector Addition on Locally Convex Space is Continuous, vector addition on $X$ is continuous.
From Scalar Multiplication on Locally Convex Space is Continuous, scalar multiplication on $X$ is continuous.
So $\struct {X, \tau}$ is a topological vector space.
{{qed}}
Category:Locally Convex Spaces
Category:Topologi... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Locally Convex Space|locally convex space]] over $\GF$ equipped with the [[Definition:Locally Convex Space/Standard Topology|standard topology]] $\tau$.
Then $\struct {X, \tau}$ is a [[Definition:Topological Vector Space|topological vector space... | From [[Vector Addition on Locally Convex Space is Continuous]], [[Definition:Vector Addition on Vector Space|vector addition]] on $X$ is [[Definition:Continuous Mapping (Topology)|continuous]].
From [[Scalar Multiplication on Locally Convex Space is Continuous]], [[Definition:Scalar Multiplication on Vector Space|sca... | Locally Convex Space is Topological Vector Space | https://proofwiki.org/wiki/Locally_Convex_Space_is_Topological_Vector_Space | https://proofwiki.org/wiki/Locally_Convex_Space_is_Topological_Vector_Space | [
"Locally Convex Space is Topological Vector Space",
"Locally Convex Spaces",
"Topological Vector Spaces",
"Locally Convex Space is Topological Vector Space"
] | [
"Definition:Locally Convex Space",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Topological Vector Space"
] | [
"Vector Addition on Locally Convex Space is Continuous",
"Definition:Vector Addition/Vector Space",
"Definition:Continuous Mapping (Topology)",
"Scalar Multiplication on Locally Convex Space is Continuous",
"Definition:Scalar Multiplication/Vector Space",
"Definition:Continuous Mapping (Topology)",
"Def... |
proofwiki-20445 | Existence of Non-Zero Continuous Linear Functional vanishing on Proper Closed Subspace of Locally Convex Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a locally convex space over $\GF$.
Let $Y$ be a proper closed linear subspace of $X$.
Let $x_0 \in X \setminus Y$.
Then there exists a continuous linear functional $f : X \to \GF$ such that:
:$\map f y = 0$ for each $y \in Y$
and:
:$\map f {x_0} \ne 0$ | Let:
:$X_0 = \map \span {Y \cup \set x}$
From Linear Span is Linear Subspace, we have:
:$X_0$ is a linear subspace of $X$.
Note that we can then write any $u \in X_0$ in the form:
:$u = y + \alpha x$
for $y \in Y$ and $\alpha \in \mathbb F$.
We want to define a map in terms of this representation, so we show that th... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Locally Convex Space|locally convex space]] over $\GF$.
Let $Y$ be a [[Definition:Proper Subset|proper]] [[Definition:Closed Linear Subspace|closed linear subspace]] of $X$.
Let $x_0 \in X \setminus Y$.
Then there exists a [[Definition:Continuo... | Let:
:$X_0 = \map \span {Y \cup \set x}$
From [[Linear Span is Linear Subspace]], we have:
:$X_0$ is a [[Definition:Linear Subspace|linear subspace]] of $X$.
Note that we can then write any $u \in X_0$ in the form:
:$u = y + \alpha x$
for $y \in Y$ and $\alpha \in \mathbb F$.
We want to define a map in terms ... | Existence of Non-Zero Continuous Linear Functional vanishing on Proper Closed Subspace of Locally Convex Space | https://proofwiki.org/wiki/Existence_of_Non-Zero_Continuous_Linear_Functional_vanishing_on_Proper_Closed_Subspace_of_Locally_Convex_Space | https://proofwiki.org/wiki/Existence_of_Non-Zero_Continuous_Linear_Functional_vanishing_on_Proper_Closed_Subspace_of_Locally_Convex_Space | [
"Linear Functionals",
"Locally Convex Spaces"
] | [
"Definition:Locally Convex Space",
"Definition:Proper Subset",
"Definition:Closed Linear Subspace",
"Definition:Continuous Mapping (Topology)",
"Definition:Linear Functional"
] | [
"Linear Span is Linear Subspace",
"Definition:Linear Subspace",
"Definition:Linear Subspace",
"Definition:Closed Set/Topology",
"Characterization of Continuous Linear Functionals on Topological Vector Space",
"Definition:Continuous Mapping (Topology)",
"Hahn-Banach Theorem for Continuous Linear Function... |
proofwiki-20446 | Equivalence of Definitions of Metrizable Space/Lemma 1 | :$d_\phi$ is a metric on $S$. | We note that by definition $d: M \to \R$ is a metric on $M$.
Hence $d$ satisfies all the metric space axioms. | :$d_\phi$ is a [[Definition:Metric|metric]] on $S$. | We note that by definition $d: M \to \R$ is a [[Definition:Metric|metric]] on $M$.
Hence $d$ satisfies all the [[Axiom:Metric Space Axioms|metric space axioms]]. | Equivalence of Definitions of Metrizable Space/Lemma 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Metrizable_Space/Lemma_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Metrizable_Space/Lemma_1 | [
"Equivalence of Definitions of Metrizable Space"
] | [
"Definition:Metric Space/Metric"
] | [
"Definition:Metric Space/Metric",
"Axiom:Metric Space Axioms",
"Definition:Metric Space/Metric"
] |
proofwiki-20447 | Equivalence of Definitions of Metrizable Space/Lemma 2 | :$\forall U \subseteq S : U$ is open in $\struct{S, d_\phi}$ {{iff}} $\phi \sqbrk U$ is open in $\struct{A, d}$ | === Lemma 3 ===
{{:Equivalence of Definitions of Metrizable Space/Lemma 3}}{{qed|lemma}}
Let $U \subseteq S$. | :$\forall U \subseteq S : U$ is [[Definition:Open Set (Metric Space)|open]] in $\struct{S, d_\phi}$ {{iff}} $\phi \sqbrk U$ is [[Definition:Open Set (Metric Space)|open]] in $\struct{A, d}$ | === [[Equivalence of Definitions of Metrizable Space/Lemma 3|Lemma 3]] ===
{{:Equivalence of Definitions of Metrizable Space/Lemma 3}}{{qed|lemma}}
Let $U \subseteq S$. | Equivalence of Definitions of Metrizable Space/Lemma 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Metrizable_Space/Lemma_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Metrizable_Space/Lemma_2 | [
"Equivalence of Definitions of Metrizable Space"
] | [
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space"
] | [
"Equivalence of Definitions of Metrizable Space/Lemma 3"
] |
proofwiki-20448 | Equivalence of Definitions of Metrizable Space/Lemma 3 | :$\forall s \in S, \epsilon \in \R_{\ge 0} : \phi \sqbrk {\map {B_\epsilon} s} = \map {B_\epsilon} {\map \phi s}$
where:
:$(1) \quad \map {B_\epsilon} s$ is the open ball in $\struct{S, d_\phi}$ with center $s$ and radius $\epsilon$
:$(2) \quad \map {B_\epsilon} {\map \phi s}$ is the open ball in $\struct{A, d}$ with c... | Let $s \in S$.
Let $\epsilon \in \R_{\ge 0}$.
We have:
{{begin-eqn}}
{{eqn | l = x \in \map {B_\epsilon} s
| o = \leadstoandfrom
| r = \map {d_\phi} {s, x} < \epsilon
| c = {{Defof|Open Ball}} in $\struct{S, d_\phi}$
}}
{{eqn | o = \leadstoandfrom
| r = \map d {\map \phi s, \map \phi x} < \epsil... | :$\forall s \in S, \epsilon \in \R_{\ge 0} : \phi \sqbrk {\map {B_\epsilon} s} = \map {B_\epsilon} {\map \phi s}$
where:
:$(1) \quad \map {B_\epsilon} s$ is the [[Definition:Open Ball|open ball]] in $\struct{S, d_\phi}$ with [[Definition:Center of Open Ball|center]] $s$ and [[Definition:Radius of Open Ball|radius]] $\e... | Let $s \in S$.
Let $\epsilon \in \R_{\ge 0}$.
We have:
{{begin-eqn}}
{{eqn | l = x \in \map {B_\epsilon} s
| o = \leadstoandfrom
| r = \map {d_\phi} {s, x} < \epsilon
| c = {{Defof|Open Ball}} in $\struct{S, d_\phi}$
}}
{{eqn | o = \leadstoandfrom
| r = \map d {\map \phi s, \map \phi x} < \ep... | Equivalence of Definitions of Metrizable Space/Lemma 3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Metrizable_Space/Lemma_3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Metrizable_Space/Lemma_3 | [
"Equivalence of Definitions of Metrizable Space"
] | [
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius"
] | [
"Definition:Set Equality",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Surjection",
"Image of Preimage of Subset under Surjection equals Subset",
"Category:Equivalence of Definitions of Metrizable Space"
] |
proofwiki-20449 | Linear Combination of Continuous Functions valued in Topological Vector Space is Continuous | Let $X$ be a topological space.
Let $K$ be a topological field.
Let $Y$ be a topological vector space over $K$.
Let $n \in \N$.
Let $f_1, \ldots, f_n : X \to Y$ be continuous functions.
Let $\alpha_1, \ldots, \alpha_n \in K$.
Then:
:$\ds \sum_{k \mathop = 1}^n \alpha_k f_k$ is a continuous function. | We do induction on the number of functions $n$: | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $K$ be a [[Definition:Topological Field|topological field]].
Let $Y$ be a [[Definition:Topological Vector Space|topological vector space]] over $K$.
Let $n \in \N$.
Let $f_1, \ldots, f_n : X \to Y$ be [[Definition:Continuous Mapping (Topology)|c... | We do [[Principle of Mathematical Induction|induction]] on the number of functions $n$: | Linear Combination of Continuous Functions valued in Topological Vector Space is Continuous | https://proofwiki.org/wiki/Linear_Combination_of_Continuous_Functions_valued_in_Topological_Vector_Space_is_Continuous | https://proofwiki.org/wiki/Linear_Combination_of_Continuous_Functions_valued_in_Topological_Vector_Space_is_Continuous | [
"Topological Vector Spaces"
] | [
"Definition:Topological Space",
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-20450 | Cartesian Space of Topological Field is Topological Vector Space | Let $K$ be a topological field.
Let $K^n$ be the cartesian $n$-th power of $K$, defining scalar multiplication and vector addition component-wise.
Equip $K^n$ with the product topology given by $K$.
Then $K^n$ is a topological vector space.
{{Refactor|level = medium|extract the following into its own (corollary) page}... | If $K$ is Hausdorff, then from Product Space is $T_2$ iff Factor Spaces are $T_2$ the topology on $K^n$ is Hausdorff.
We therefore just need to show that scalar multiplication and vector addition are continuous. | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $K^n$ be the [[Definition:Cartesian Space|cartesian $n$-th power of $K$]], defining [[Definition:Scalar Multiplication on Vector Space|scalar multiplication]] and [[Definition:Vector Addition on Vector Space|vector addition]] component-wise.
Equip $... | If $K$ is [[Definition:Hausdorff Space|Hausdorff]], then from [[Product Space is T2 iff Factor Spaces are T2|Product Space is $T_2$ iff Factor Spaces are $T_2$]] the [[Definition:Topology|topology]] on $K^n$ is [[Definition:Hausdorff Space|Hausdorff]].
We therefore just need to show that [[Definition:Scalar Multiplica... | Cartesian Space of Topological Field is Topological Vector Space | https://proofwiki.org/wiki/Cartesian_Space_of_Topological_Field_is_Topological_Vector_Space | https://proofwiki.org/wiki/Cartesian_Space_of_Topological_Field_is_Topological_Vector_Space | [
"Topological Vector Spaces",
"Topological Fields",
"Hausdorff Topological Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Cartesian Product/Cartesian Space",
"Definition:Scalar Multiplication/Vector Space",
"Definition:Vector Addition/Vector Space",
"Definition:Product Topology",
"Definition:Topological Vector Space",
"Definition:T2 Space",
"Definition:Topological Field",
"De... | [
"Definition:T2 Space",
"Product Space is T2 iff Factor Spaces are T2",
"Definition:Topology",
"Definition:T2 Space",
"Definition:Scalar Multiplication/Vector Space",
"Definition:Vector Addition/Vector Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"De... |
proofwiki-20451 | Sum of Continuous Functions on Topological Ring is Continuous | Let $X$ be a topological space.
Let $R$ be a topological ring.
Let $f, g : X \to R$ be continuous mappings.
Then $f + g : X \to R$ is continuous. | Equip the Cartesian product $R \times R$ with its product topology.
Define $h : X \to R \times R$ by:
:$\map h x = \tuple {\map f x, \map g x}$
for each $x \in X$ and $s : R \times R \to R$ by:
:$\map s {x, y} = x + y$
for each $\tuple {x, y} \in R \times R$.
From Continuous Mapping to Product Space, $h$ is contin... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $R$ be a [[Definition:Topological Ring|topological ring]].
Let $f, g : X \to R$ be [[Definition:Continuous Mapping (Topology)|continuous mappings]].
Then $f + g : X \to R$ is [[Definition:Continuous Mapping (Topology)|continuous]]. | Equip the [[Definition:Cartesian Product|Cartesian product]] $R \times R$ with its [[Definition:Product Topology|product topology]].
Define $h : X \to R \times R$ by:
:$\map h x = \tuple {\map f x, \map g x}$
for each $x \in X$ and $s : R \times R \to R$ by:
:$\map s {x, y} = x + y$
for each $\tuple {x, y} \in... | Sum of Continuous Functions on Topological Ring is Continuous | https://proofwiki.org/wiki/Sum_of_Continuous_Functions_on_Topological_Ring_is_Continuous | https://proofwiki.org/wiki/Sum_of_Continuous_Functions_on_Topological_Ring_is_Continuous | [
"Continuous Mappings (Topology)",
"Topological Rings"
] | [
"Definition:Topological Space",
"Definition:Topological Ring",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Cartesian Product",
"Definition:Product Topology",
"Continuous Mapping to Product Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Topological Ring",
"Definition:Continuous Mapping (Topology)",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Mapping ... |
proofwiki-20452 | Product of Continuous Functions on Topological Ring is Continuous | Let $X$ be a topological space.
Let $R$ be a topological ring.
Let $f, g : X \to R$ be continuous functions.
Then $f \cdot g : X \to R$ is continuous. | Equip the Cartesian product $R \times R$ with its product topology.
Define $h : X \to R \times R$ by:
:$\map h x = \tuple {\map f x, \map g x}$
for each $x \in X$ and $p : R \times R \to R$ by:
:$\map p {x, y} = x y$
for each $\tuple {x, y} \in R \times R$.
From Continuous Mapping to Product Space, $h$ is continuous... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $R$ be a [[Definition:Topological Ring|topological ring]].
Let $f, g : X \to R$ be [[Definition:Continuous Mapping (Topology)|continuous functions]].
Then $f \cdot g : X \to R$ is [[Definition:Continuous Mapping (Topology)|continuous]]. | Equip the [[Definition:Cartesian Product|Cartesian product]] $R \times R$ with its [[Definition:Product Topology|product topology]].
Define $h : X \to R \times R$ by:
:$\map h x = \tuple {\map f x, \map g x}$
for each $x \in X$ and $p : R \times R \to R$ by:
:$\map p {x, y} = x y$
for each $\tuple {x, y} \in R ... | Product of Continuous Functions on Topological Ring is Continuous | https://proofwiki.org/wiki/Product_of_Continuous_Functions_on_Topological_Ring_is_Continuous | https://proofwiki.org/wiki/Product_of_Continuous_Functions_on_Topological_Ring_is_Continuous | [
"Topological Rings"
] | [
"Definition:Topological Space",
"Definition:Topological Ring",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Cartesian Product",
"Definition:Product Topology",
"Continuous Mapping to Product Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Topological Ring",
"Definition:Continuous Mapping (Topology)",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Mapping ... |
proofwiki-20453 | Normed Dual Space of Normed Quotient Vector Space is Isometrically Isomorphic to Annihilator | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.
Let $X^\ast$ be the normed dual space of $X$.
Let $Y$ be a closed linear subspace of $X$.
Let $\paren {X/Y}^\ast$ be the normed dual space of the normed quotient vector space $X/Y$.
Then:
:$\paren {X/Y}^\ast$ is... | Let $\pi$ be the quotient mapping associated with $X/Y$.
Note that for each $f \in \paren {X/Y}^\ast$ we have:
:$f \circ \pi \in X^\ast$
by Composition of Bounded Linear Transformations is Bounded Linear Transformation.
Further, for $y \in Y$ we have $\map \pi y = 0_{X/Y}$ from Kernel of Quotient Mapping, and so:
:$... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Let $Y$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $X$.
Let $\paren ... | Let $\pi$ be the [[Definition:Quotient Mapping|quotient mapping]] associated with $X/Y$.
Note that for each $f \in \paren {X/Y}^\ast$ we have:
:$f \circ \pi \in X^\ast$
by [[Composition of Bounded Linear Transformations is Bounded Linear Transformation]].
Further, for $y \in Y$ we have $\map \pi y = 0_{X/Y}$ from... | Normed Dual Space of Normed Quotient Vector Space is Isometrically Isomorphic to Annihilator | https://proofwiki.org/wiki/Normed_Dual_Space_of_Normed_Quotient_Vector_Space_is_Isometrically_Isomorphic_to_Annihilator | https://proofwiki.org/wiki/Normed_Dual_Space_of_Normed_Quotient_Vector_Space_is_Isometrically_Isomorphic_to_Annihilator | [
"Annihilators of Subspaces of Banach Spaces",
"Normed Dual Spaces",
"Normed Quotient Vector Spaces",
"Annihilators of Subspaces of Banach Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Closed Linear Subspace",
"Definition:Normed Dual Space",
"Definition:Normed Quotient Vector Space",
"Definition:Isometric Isomorphism",
"Definition:Annihilator of Subspace of Banach Space"
] | [
"Definition:Quotient Mapping",
"Norm on Bounded Linear Transformation is Submultiplicative",
"Kernel of Quotient Mapping",
"Linear Isometry is Injective/Corollary",
"Definition:Linear Isometry",
"Definition:Surjection",
"Definition:Linear Transformation",
"Quotient Mapping Maps Unit Open Ball in Norme... |
proofwiki-20454 | Double Angle Formula for Tangent/Corollary | Let $u = \tan \dfrac \theta 2$.
Then:
:$\tan \theta = \dfrac {2 u} {1 - u^2}$ | From Double Angle Formula for Tangent:
:$\tan 2 \theta = \dfrac {2 \tan \theta} {1 - \tan^2 \theta}$
The result follows by substituting $\dfrac \theta 2$ for $\theta$.
{{qed}} | Let $u = \tan \dfrac \theta 2$.
Then:
:$\tan \theta = \dfrac {2 u} {1 - u^2}$ | From [[Double Angle Formula for Tangent]]:
:$\tan 2 \theta = \dfrac {2 \tan \theta} {1 - \tan^2 \theta}$
The result follows by substituting $\dfrac \theta 2$ for $\theta$.
{{qed}} | Double Angle Formula for Tangent/Corollary | https://proofwiki.org/wiki/Double_Angle_Formula_for_Tangent/Corollary | https://proofwiki.org/wiki/Double_Angle_Formula_for_Tangent/Corollary | [
"Double Angle Formula for Tangent"
] | [] | [
"Double Angle Formulas/Tangent"
] |
proofwiki-20455 | Linear Transformation from Cartesian Space on Hausdorff Topological Field to Topological Vector Space is Continuous | Let $K$ be a topological field.
Let $K^n$ be the $n$-Cartesian space as a topological vector space with its product topology.
Let $X$ be a topological vector space over $K$.
Let $f : K^n \to X$ be a linear transformation.
Then $f$ is continuous. | Let $\set {e_1, \ldots, e_n}$ be the standard basis for $K^n$.
For each $\alpha_1, \alpha_2, \ldots, \alpha_n$ we have:
:$\ds \map f {\sum_{i \mathop = 1}^n \alpha_i e_i} = \sum_{i \mathop = 1}^n \alpha_i \map f {e_i}$
Let $\pr_i : K^n \to K$ be the projection of $K^n$ onto its $i$th factor, so that:
:$\ds \map {\pr_... | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $K^n$ be the [[Definition:Cartesian Space|$n$-Cartesian space]] as a [[Definition:Topological Vector Space|topological vector space]] with its [[Definition:Product Topology|product topology]].
Let $X$ be a [[Definition:Topological Vector Space|topol... | Let $\set {e_1, \ldots, e_n}$ be the standard [[Definition:Basis of Vector Space|basis]] for $K^n$.
For each $\alpha_1, \alpha_2, \ldots, \alpha_n$ we have:
:$\ds \map f {\sum_{i \mathop = 1}^n \alpha_i e_i} = \sum_{i \mathop = 1}^n \alpha_i \map f {e_i}$
Let $\pr_i : K^n \to K$ be the [[Definition:Projection (Mapp... | Linear Transformation from Cartesian Space on Hausdorff Topological Field to Topological Vector Space is Continuous | https://proofwiki.org/wiki/Linear_Transformation_from_Cartesian_Space_on_Hausdorff_Topological_Field_to_Topological_Vector_Space_is_Continuous | https://proofwiki.org/wiki/Linear_Transformation_from_Cartesian_Space_on_Hausdorff_Topological_Field_to_Topological_Vector_Space_is_Continuous | [
"Topological Fields",
"Topological Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Cartesian Product/Cartesian Space",
"Definition:Topological Vector Space",
"Definition:Product Topology",
"Definition:Topological Vector Space",
"Definition:Linear Transformation",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Basis of Vector Space",
"Definition:Projection (Mapping Theory)",
"Definition:Product Topology/Factor Space",
"Projection from Product Topology is Continuous",
"Definition:Continuous Mapping (Topology)",
"Linear Combination of Continuous Functions valued in Topological Vector Space is Continuo... |
proofwiki-20456 | Balanced Set in Topological Vector Space is Connected | Let $\GF \in \set {\R, \C}$.
Let $X$ be a topological vector space over $\GF$.
Let $B \subseteq X$ be a balanced set.
Then $B$ is connected. | We first show that $B$ is path connected.
Let $x \in B$.
Define $p : \closedint 0 1 \to X$ by:
:$\map p t = t x$
We clearly have:
:$\map p 0 = 0$
and:
:$\map p 1 = x$
Since $B$ is balanced, we have:
:$t x \in B$
for each $t \in \closedint 0 1$.
It remains to show that $p$ is continuous.
Define $m : \closedint 0 1 ... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $B \subseteq X$ be a [[Definition:Balanced Set|balanced set]].
Then $B$ is [[Definition:Connected Set (Topology)|connected]]. | We first show that $B$ is [[Definition:Path-Connected Space|path connected]].
Let $x \in B$.
Define $p : \closedint 0 1 \to X$ by:
:$\map p t = t x$
We clearly have:
:$\map p 0 = 0$
and:
:$\map p 1 = x$
Since $B$ is [[Definition:Balanced Set|balanced]], we have:
:$t x \in B$
for each $t \in \closedint 0 ... | Balanced Set in Topological Vector Space is Connected | https://proofwiki.org/wiki/Balanced_Set_in_Topological_Vector_Space_is_Connected | https://proofwiki.org/wiki/Balanced_Set_in_Topological_Vector_Space_is_Connected | [
"Balanced Sets",
"Connected Topological Spaces",
"Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Balanced Set",
"Definition:Connected Set (Topology)"
] | [
"Definition:Path-Connected/Topological Space",
"Definition:Balanced Set",
"Definition:Continuous Mapping (Topology)",
"Restriction of Continuous Mapping is Continuous",
"Definition:Continuous Mapping (Topology)",
"Definition:Horizontal Section of Function",
"Horizontal Section of Continuous Function is ... |
proofwiki-20457 | Isomorphism from Cartesian Space to Finite-Dimensional Subspace of Hausdorff Topological Vector Space is Homeomorphism | Let $\GF \in \set {\R, \C}$.
Let $X$ be a Hausdorff topological vector space over $\GF$.
Let $n \in \N$.
Let $Y$ be a subspace of $X$ with dimension $n$.
Let $f : \GF^n \to Y$ be a vector space isomorphism.
Then $f$ is a homeomorphism. | First, we assure ourselves that such a vector space isomorphism $f : \GF^n \to Y$ exists from Same Dimensional Vector Spaces are Isomorphic.
From Linear Transformation from Cartesian Space on Hausdorff Topological Field to Topological Vector Space is Continuous, $f$ is continuous.
It suffices to show that $f^{-1}$ is c... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Hausdorff Topological Vector Space|Hausdorff topological vector space]] over $\GF$.
Let $n \in \N$.
Let $Y$ be a [[Definition:Linear Subspace|subspace]] of $X$ with [[Definition:Dimension of Vector Space|dimension]] $n$.
Let $f : \GF^n \to Y$ be a [[Definiti... | First, we assure ourselves that such a [[Definition:Vector Space Isomorphism|vector space isomorphism]] $f : \GF^n \to Y$ exists from [[Same Dimensional Vector Spaces are Isomorphic]].
From [[Linear Transformation from Cartesian Space on Hausdorff Topological Field to Topological Vector Space is Continuous]], $f$ is [... | Isomorphism from Cartesian Space to Finite-Dimensional Subspace of Hausdorff Topological Vector Space is Homeomorphism | https://proofwiki.org/wiki/Isomorphism_from_Cartesian_Space_to_Finite-Dimensional_Subspace_of_Hausdorff_Topological_Vector_Space_is_Homeomorphism | https://proofwiki.org/wiki/Isomorphism_from_Cartesian_Space_to_Finite-Dimensional_Subspace_of_Hausdorff_Topological_Vector_Space_is_Homeomorphism | [
"Topological Vector Spaces"
] | [
"Definition:Hausdorff Topological Vector Space",
"Definition:Linear Subspace",
"Definition:Dimension of Vector Space",
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism",
"Definition:Homeomorphism"
] | [
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism",
"Same Dimensional Vector Spaces are Isomorphic",
"Linear Transformation from Cartesian Space on Hausdorff Topological Field to Topological Vector Space is Continuous",
"Definition:Continuous Mapping (Topolo... |
proofwiki-20458 | Finite-Dimensional Subspace of Hausdorff Topological Vector Space is Closed | Let $\GF \in \set {\R, \C}$.
Let $X$ be a Hausdorff topological vector space over $\GF$.
Let $n \in \N$.
Let $Y$ be a subspace of $X$ with dimension $n$.
Then $Y$ is closed. | Let $f : \GF^n \to Y$ be a vector space isomorphism.
From Isomorphism from Cartesian Space to Finite-Dimensional Subspace of Hausdorff Topological Vector Space is Homeomorphism, $f$ is a homeomorphism.
Let $B$ be the unit ball in $\GF^n$.
Let $D$ be the closed unit ball in $\GF^n$.
Let $S$ be the unit sphere in $\GF... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Hausdorff Topological Vector Space|Hausdorff topological vector space]] over $\GF$.
Let $n \in \N$.
Let $Y$ be a [[Definition:Linear Subspace|subspace]] of $X$ with [[Definition:Dimension of Vector Space|dimension]] $n$.
Then $Y$ is [[Definition:Closed Set|... | Let $f : \GF^n \to Y$ be a [[Definition:Vector Space Isomorphism|vector space isomorphism]].
From [[Isomorphism from Cartesian Space to Finite-Dimensional Subspace of Hausdorff Topological Vector Space is Homeomorphism]], $f$ is a [[Definition:Homeomorphism|homeomorphism]].
Let $B$ be the [[Definition:Unit Ball|unit... | Finite-Dimensional Subspace of Hausdorff Topological Vector Space is Closed | https://proofwiki.org/wiki/Finite-Dimensional_Subspace_of_Hausdorff_Topological_Vector_Space_is_Closed | https://proofwiki.org/wiki/Finite-Dimensional_Subspace_of_Hausdorff_Topological_Vector_Space_is_Closed | [
"Hausdorff Topological Vector Spaces"
] | [
"Definition:Hausdorff Topological Vector Space",
"Definition:Linear Subspace",
"Definition:Dimension of Vector Space",
"Definition:Closed Set"
] | [
"Definition:Isomorphism (Abstract Algebra)/R-Algebraic Structure Isomorphism/Vector Space Isomorphism",
"Isomorphism from Cartesian Space to Finite-Dimensional Subspace of Hausdorff Topological Vector Space is Homeomorphism",
"Definition:Homeomorphism",
"Definition:Unit Ball",
"Definition:Closed Unit Ball",... |
proofwiki-20459 | Dilation of Compact Set in Topological Vector Space is Compact | Let $k$ be a topological field.
Let $X$ be a topological vector space over $X$.
Let $K$ be a compact subset of $X$.
Let $t \in k \setminus \set {0_k}$.
Then $t K$ is compact. | Let $\family {U_\alpha : \alpha \in I}$ be open sets such that:
:$\ds t K \subseteq \bigcup_{\alpha \mathop \in I} U_\alpha$
From Dilation of Union of Subsets of Vector Space, we have:
:$\ds K \subseteq \bigcup_{\alpha \mathop \in I} \paren {t^{-1} U_\alpha}$
From Dilation of Open Set in Topological Vector Space is ... | Let $k$ be a [[Definition:Topological Field|topological field]].
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $X$.
Let $K$ be a [[Definition:Compact Topological Space|compact subset]] of $X$.
Let $t \in k \setminus \set {0_k}$.
Then $t K$ is [[Definition:Compact Topological... | Let $\family {U_\alpha : \alpha \in I}$ be [[Definition:Open Set|open sets]] such that:
:$\ds t K \subseteq \bigcup_{\alpha \mathop \in I} U_\alpha$
From [[Dilation of Union of Subsets of Vector Space]], we have:
:$\ds K \subseteq \bigcup_{\alpha \mathop \in I} \paren {t^{-1} U_\alpha}$
From [[Dilation of Open Set... | Dilation of Compact Set in Topological Vector Space is Compact/Proof 1 | https://proofwiki.org/wiki/Dilation_of_Compact_Set_in_Topological_Vector_Space_is_Compact | https://proofwiki.org/wiki/Dilation_of_Compact_Set_in_Topological_Vector_Space_is_Compact/Proof_1 | [
"Dilation of Compact Set in Topological Vector Space is Compact",
"Dilations of Subsets of Vector Spaces",
"Topological Vector Spaces",
"Compact Topological Spaces"
] | [
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space"
] | [
"Definition:Open Set",
"Dilation of Union of Subsets of Vector Space",
"Dilation of Open Set in Topological Vector Space is Open",
"Definition:Open Set",
"Definition:Compact Topological Space",
"Dilation of Union of Subsets of Vector Space",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"De... |
proofwiki-20460 | Dilation of Compact Set in Topological Vector Space is Compact | Let $k$ be a topological field.
Let $X$ be a topological vector space over $X$.
Let $K$ be a compact subset of $X$.
Let $t \in k \setminus \set {0_k}$.
Then $t K$ is compact. | From Dilation Mapping on Topological Vector Space is Continuous, the mapping $c_t : X \to X$ defined by:
:$\map {c_t} x = t x$
for each $x \in X$ is continuous.
From Continuous Image of Compact Space is Compact:
:$\map {c_t} K = t K$ is compact.
{{qed}} | Let $k$ be a [[Definition:Topological Field|topological field]].
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $X$.
Let $K$ be a [[Definition:Compact Topological Space|compact subset]] of $X$.
Let $t \in k \setminus \set {0_k}$.
Then $t K$ is [[Definition:Compact Topological... | From [[Dilation Mapping on Topological Vector Space is Continuous]], the [[Definition:Mapping|mapping]] $c_t : X \to X$ defined by:
:$\map {c_t} x = t x$
for each $x \in X$ is [[Definition:Continuous Mapping|continuous]].
From [[Continuous Image of Compact Space is Compact]]:
:$\map {c_t} K = t K$ is [[Definition:Comp... | Dilation of Compact Set in Topological Vector Space is Compact/Proof 2 | https://proofwiki.org/wiki/Dilation_of_Compact_Set_in_Topological_Vector_Space_is_Compact | https://proofwiki.org/wiki/Dilation_of_Compact_Set_in_Topological_Vector_Space_is_Compact/Proof_2 | [
"Dilation of Compact Set in Topological Vector Space is Compact",
"Dilations of Subsets of Vector Spaces",
"Topological Vector Spaces",
"Compact Topological Spaces"
] | [
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space"
] | [
"Dilation Mapping on Topological Vector Space is Continuous",
"Definition:Mapping",
"Definition:Continuous Mapping",
"Continuous Image of Compact Space is Compact",
"Definition:Compact Topological Space"
] |
proofwiki-20461 | Topological Dual Space of Hausdorff Locally Convex Space Separates Points | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\GF$.
Let $X^\ast$ be the topological dual of $X$.
Let $x \in X$.
Then $X^\ast$ separates points.
That is, if $x, y \in X$ are such that:
:$\map f x = \map f y$ for all $f \in X^\ast$
we have that $x = y$. | Suppose $x \ne y$.
It suffices to find $f \in X^\ast$ such that $\map f x \ne \map f y$.
From Finite Topological Space is Compact, $\set {\mathbf 0_X}$ is compact.
From Compact Subspace of Hausdorff Space is Closed, $\set {\mathbf 0_X}$ is closed.
Since $x \ne y$, we have:
:$x - y \not \in X \setminus \set {\mathbf... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Locally Convex Space/Hausdorff|Hausdorff]] [[Definition:Locally Convex Space|locally convex space]] over $\GF$.
Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual]] of $X$.
Let $x \in X$.
Then $X^\ast$ [[Definition:Mappi... | Suppose $x \ne y$.
It suffices to find $f \in X^\ast$ such that $\map f x \ne \map f y$.
From [[Finite Topological Space is Compact]], $\set {\mathbf 0_X}$ is [[Definition:Compact Topological Space|compact]].
From [[Compact Subspace of Hausdorff Space is Closed]], $\set {\mathbf 0_X}$ is [[Definition:Closed Set|c... | Topological Dual Space of Hausdorff Locally Convex Space Separates Points | https://proofwiki.org/wiki/Topological_Dual_Space_of_Hausdorff_Locally_Convex_Space_Separates_Points | https://proofwiki.org/wiki/Topological_Dual_Space_of_Hausdorff_Locally_Convex_Space_Separates_Points | [
"Locally Convex Spaces",
"Topological Dual Spaces"
] | [
"Definition:Locally Convex Space/Hausdorff",
"Definition:Locally Convex Space",
"Definition:Topological Dual Space",
"Definition:Mappings Separating Points"
] | [
"Finite Topological Space is Compact",
"Definition:Compact Topological Space",
"Compact Subspace of Hausdorff Space is Closed",
"Definition:Closed Set",
"Existence of Non-Zero Continuous Linear Functional vanishing on Proper Closed Subspace of Locally Convex Space",
"Category:Locally Convex Spaces",
"Ca... |
proofwiki-20462 | Characterization of von Neumann-Boundedness in Hausdorff Locally Convex Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\GF$.
Let $U \subseteq X$.
Then $U$ is von Neumann-bounded {{iff}}:
:for each $p \in \PP$ there exists $C_p > 0$ such that:
::$\map p x < C_p$
:for each $x \in U$. | For each $p \in \PP$, let:
:$B_p = \set {y \in X : \map p y < 1}$
Note that by the definition of the standard topology, $B_p$ is an open neighborhood of $\mathbf 0_X$.
Let $r > 0$.
From {{SeminormAxiom|2}}, for $y \in X$ we have:
:$\map p y < 1$
{{iff}}:
:$\map p {r y} < r$
So, we have:
:$r B_p = \set {y \in X :... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Locally Convex Space/Hausdorff|Hausdorff]] [[Definition:Locally Convex Space|locally convex space]] over $\GF$.
Let $U \subseteq X$.
Then $U$ is [[Definition:von Neumann-Bounded Subset of Topological Vector Space|von Neumann-bounded]] {{iff}}:
... | For each $p \in \PP$, let:
:$B_p = \set {y \in X : \map p y < 1}$
Note that by the definition of the [[Definition:Locally Convex Space/Standard Topology|standard topology]], $B_p$ is an [[Definition:Open Neighborhood|open neighborhood]] of $\mathbf 0_X$.
Let $r > 0$.
From {{SeminormAxiom|2}}, for $y \in X$ we ha... | Characterization of von Neumann-Boundedness in Hausdorff Locally Convex Space | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Boundedness_in_Hausdorff_Locally_Convex_Space | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Boundedness_in_Hausdorff_Locally_Convex_Space | [
"Von Neumann-Bounded Subsets of Topological Vector Spaces",
"Locally Convex Spaces"
] | [
"Definition:Locally Convex Space/Hausdorff",
"Definition:Locally Convex Space",
"Definition:von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Definition:Locally Convex Space/Standard Topology",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood"
] |
proofwiki-20463 | Normed Vector Space is Hausdorff Locally Convex Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Then $\struct {X, \norm {\, \cdot \,} }$ is a Hausdorff locally convex space. | From Normed Vector Space is Locally Convex Space, $\struct {X, \norm {\, \cdot \,} }$ can be viewed as the locally convex space $\struct {X, \norm {\, \cdot \,} }$.
Now for $x \ne \mathbf 0_X$, we have:
:$\norm x \ne 0$
from {{NormAxiomVector|1}}.
So $\struct {X, \norm {\, \cdot \,} }$ is a Hausdorff locally convex ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Then $\struct {X, \norm {\, \cdot \,} }$ is a [[Definition:Locally Convex Space/Hausdorff|Hausdorff]] [[Definition:Locally Convex Space|locally convex space]]. | From [[Normed Vector Space is Locally Convex Space]], $\struct {X, \norm {\, \cdot \,} }$ can be viewed as the [[Definition:Locally Convex Space|locally convex space]] $\struct {X, \norm {\, \cdot \,} }$.
Now for $x \ne \mathbf 0_X$, we have:
:$\norm x \ne 0$
from {{NormAxiomVector|1}}.
So $\struct {X, \norm {\,... | Normed Vector Space is Hausdorff Locally Convex Space | https://proofwiki.org/wiki/Normed_Vector_Space_is_Hausdorff_Locally_Convex_Space | https://proofwiki.org/wiki/Normed_Vector_Space_is_Hausdorff_Locally_Convex_Space | [
"Locally Convex Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Locally Convex Space/Hausdorff",
"Definition:Locally Convex Space"
] | [
"Normed Vector Space is Locally Convex Space",
"Definition:Locally Convex Space",
"Definition:Locally Convex Space/Hausdorff",
"Definition:Locally Convex Space",
"Category:Locally Convex Spaces",
"Category:Normed Vector Spaces"
] |
proofwiki-20464 | Characterization of von Neumann-Boundedness in Normed Vector Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $U \subseteq X$.
Then $U$ is von Neumann-bounded {{iff}} it is bounded as a subset of a normed vector space. | From Normed Vector Space is Hausdorff Locally Convex Space, $\struct {X, \norm {\, \cdot \,} }$ can be viewed as the Hausdorff locally convex space $\struct {X, \norm {\, \cdot \,} }$.
The result is then immediate from Characterization of von Neumann-Boundedness in Hausdorff Locally Convex Space.
{{qed}}
Category:Von ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $U \subseteq X$.
Then $U$ is [[Definition:von Neumann-Bounded Subset of Topological Vector Space|von Neumann-bounded]] {{iff}} it is [[Definition:Bounded Subset of Normed Vector Spa... | From [[Normed Vector Space is Hausdorff Locally Convex Space]], $\struct {X, \norm {\, \cdot \,} }$ can be viewed as the [[Definition:Locally Convex Space/Hausdorff|Hausdorff]] [[Definition:Locally Convex Space|locally convex space]] $\struct {X, \norm {\, \cdot \,} }$.
The result is then immediate from [[Characteriz... | Characterization of von Neumann-Boundedness in Normed Vector Space | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Boundedness_in_Normed_Vector_Space | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Boundedness_in_Normed_Vector_Space | [
"Von Neumann-Bounded Subsets of Topological Vector Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:von Neumann-Bounded Subset of Topological Vector Space",
"Definition:Bounded Subset of Normed Vector Space"
] | [
"Normed Vector Space is Hausdorff Locally Convex Space",
"Definition:Locally Convex Space/Hausdorff",
"Definition:Locally Convex Space",
"Characterization of von Neumann-Boundedness in Hausdorff Locally Convex Space",
"Category:Von Neumann-Bounded Subsets of Topological Vector Spaces",
"Category:Normed Ve... |
proofwiki-20465 | Initial Topology on Normed Vector Space is Weak Topology | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.
Let $X^\ast$ be the normed dual space of $X$.
Let $w$ be the initial topology on $X$ with respect to $X^\ast$.
Then $w$ is the weak topology on $X$. | From Normed Dual Space Separates Points, if $x \ne y$ then there exists $f \in X^\ast$ such that $\map f x \ne \map f y$.
That is, if $x \ne \mathbf 0_X$, there exists $f \in X^\ast$ such that $\map f x \ne 0$.
{{qed}}
Category:Weak Topologies on Topological Vector Spaces
Category:Normed Vector Spaces
fs598imvc8zv7xhcr... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Let $w$ be the [[Definition:Initial Topology|initial topology]] on $X$ with respect to $X^\ast... | From [[Normed Dual Space Separates Points]], if $x \ne y$ then there exists $f \in X^\ast$ such that $\map f x \ne \map f y$.
That is, if $x \ne \mathbf 0_X$, there exists $f \in X^\ast$ such that $\map f x \ne 0$.
{{qed}}
[[Category:Weak Topologies on Topological Vector Spaces]]
[[Category:Normed Vector Spaces]]
fs5... | Initial Topology on Normed Vector Space is Weak Topology | https://proofwiki.org/wiki/Initial_Topology_on_Normed_Vector_Space_is_Weak_Topology | https://proofwiki.org/wiki/Initial_Topology_on_Normed_Vector_Space_is_Weak_Topology | [
"Weak Topologies on Topological Vector Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Initial Topology",
"Definition:Weak Topology on Topological Vector Space"
] | [
"Normed Dual Space Separates Points",
"Category:Weak Topologies on Topological Vector Spaces",
"Category:Normed Vector Spaces"
] |
proofwiki-20466 | Weak Topology on Topological Vector Space over Hausdorff Topological Field is Hausdorff | Let $K$ be a Hausdorff topological field.
Let $X$ be a topological vector space over $K$ with weak topology $w$.
Then $\struct {X, w}$ is Hausdorff. | From the definition of the weak topology, if $x, y \in X$ have $x \ne y$, then $x - y \ne \mathbf 0_X$:
:there exists $f \in X^\ast$ such that $\map f {x - y} \ne 0$.
Since $f$ is linear, we then have $\map f x \ne \map f y$.
Since $K$ is Hausdorff, we therefore obtain that $\struct {X, w}$ is Hausdorff from Initial ... | Let $K$ be a [[Definition:Hausdorff Space|Hausdorff]] [[Definition:Topological Field|topological field]].
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $K$ with [[Definition:Weak Topology on Topological Vector Space|weak topology]] $w$.
Then $\struct {X, w}$ is [[Definition:Haus... | From the definition of the [[Definition:Weak Topology on Topological Vector Space|weak topology]], if $x, y \in X$ have $x \ne y$, then $x - y \ne \mathbf 0_X$:
:there exists $f \in X^\ast$ such that $\map f {x - y} \ne 0$.
Since $f$ is [[Definition:Linear Functional|linear]], we then have $\map f x \ne \map f y$.
... | Weak Topology on Topological Vector Space over Hausdorff Topological Field is Hausdorff | https://proofwiki.org/wiki/Weak_Topology_on_Topological_Vector_Space_over_Hausdorff_Topological_Field_is_Hausdorff | https://proofwiki.org/wiki/Weak_Topology_on_Topological_Vector_Space_over_Hausdorff_Topological_Field_is_Hausdorff | [
"Weak Topologies on Topological Vector Spaces",
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:T2 Space"
] | [
"Definition:Weak Topology on Topological Vector Space",
"Definition:Linear Functional",
"Definition:T2 Space",
"Definition:T2 Space",
"Initial Topology with respect to Point-Separating Family of Mappings onto Hausdorff Spaces is Hausdorff",
"Category:Weak Topologies on Topological Vector Spaces",
"Categ... |
proofwiki-20467 | Equation of Wave with Constant Velocity | Let $\phi$ be a wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$ and without change of shape.
Let $\paren {\map \phi x}_{t \mathop = 0} = \map f x$ be the '''wave profile''' of $\phi$.
Then the '''disturbance''' of $\phi$ at point $x$ and time $t$ can be expressed using t... | Let us imagine a snapshot of $\phi$ at the time $t = 0$.
Then, {{hypothesis}}, the wave $\phi$ is described by the equation:
:$\phi = \map f x$
Also {{hypothesis}}, $\phi$ is propagated with no change of shape.
Hence, an imagined snapshot of $\phi$ at the general time $t$ will be:
:identical with that at $t = 0$
:moved... | Let $\phi$ be a [[Definition:Wave|wave]] which is [[Definition:Direction of Propagation of Wave|propagated]] along the [[Definition:X-Axis|$x$-axis]] in the [[Definition:Positive Direction|positive direction]] with [[Definition:Constant|constant]] [[Definition:Velocity|velocity]] $c$ and without change of shape.
Let $... | Let us imagine a snapshot of $\phi$ at the [[Definition:Time|time]] $t = 0$.
Then, {{hypothesis}}, the [[Definition:Wave|wave]] $\phi$ is described by the equation:
:$\phi = \map f x$
Also {{hypothesis}}, $\phi$ is [[Definition:Direction of Propagation of Wave|propagated]] with no change of shape.
Hence, an imagined... | Equation of Wave with Constant Velocity | https://proofwiki.org/wiki/Equation_of_Wave_with_Constant_Velocity | https://proofwiki.org/wiki/Equation_of_Wave_with_Constant_Velocity | [
"Equation of Wave with Constant Velocity",
"Examples of Use of Wave Equation"
] | [
"Definition:Wave",
"Definition:Wave/Direction of Propagation",
"Definition:Axis/X-Axis",
"Definition:Axis/Positive Direction",
"Definition:Constant",
"Definition:Velocity",
"Definition:Wave Profile",
"Definition:Wave/Disturbance",
"Definition:Point",
"Definition:Instant of Time",
"Definition:Equ... | [
"Definition:Time",
"Definition:Wave",
"Definition:Wave/Direction of Propagation",
"Definition:Time",
"Definition:Axis/X-Axis",
"Definition:Distance between Points",
"Equations of Motion with Constant Acceleration/Distance after Time",
"Definition:Axis/Positive Direction",
"Definition:Axis/X-Axis",
... |
proofwiki-20468 | Equation of Wave with Constant Velocity/Corollary | Let $\phi$ be a wave which is propagated along the $x$-axis in the negative direction with constant velocity $c$ and without change of shape.
Let $\paren {\map \phi x}_{t \mathop = 0} = \map f x$ be the '''wave profile''' of $\phi$.
Then the '''disturbance''' of $\phi$ at point $x$ and time $t$ can be expressed using t... | We have {{hypothesis}} that the velocity of $\phi$ in the negative direction is $c$.
Hence the velocity of $\phi$ in the positive direction is $-c$.
By Equation of Wave with Constant Velocity:
:$\phi = \map f {x - \paren {-c} t}$
Hence the result.
{{qed}} | Let $\phi$ be a [[Definition:Wave|wave]] which is [[Definition:Direction of Propagation of Wave|propagated]] along the [[Definition:X-Axis|$x$-axis]] in the [[Definition:Negative Direction|negative direction]] with [[Definition:Constant|constant]] [[Definition:Velocity|velocity]] $c$ and without change of shape.
Let $... | We have {{hypothesis}} that the [[Definition:Velocity|velocity]] of $\phi$ in the [[Definition:Negative Direction|negative direction]] is $c$.
Hence the [[Definition:Velocity|velocity]] of $\phi$ in the [[Definition:Positive Direction|positive direction]] is $-c$.
By [[Equation of Wave with Constant Velocity]]:
:$\ph... | Equation of Wave with Constant Velocity/Corollary | https://proofwiki.org/wiki/Equation_of_Wave_with_Constant_Velocity/Corollary | https://proofwiki.org/wiki/Equation_of_Wave_with_Constant_Velocity/Corollary | [
"Equation of Wave with Constant Velocity"
] | [
"Definition:Wave",
"Definition:Wave/Direction of Propagation",
"Definition:Axis/X-Axis",
"Definition:Axis/Negative Direction",
"Definition:Constant",
"Definition:Velocity",
"Definition:Wave Profile",
"Definition:Wave/Disturbance",
"Definition:Point",
"Definition:Instant of Time",
"Definition:Equ... | [
"Definition:Velocity",
"Definition:Axis/Negative Direction",
"Definition:Velocity",
"Definition:Axis/Positive Direction",
"Equation of Wave with Constant Velocity"
] |
proofwiki-20469 | Characterization of Continuity of Linear Functional in Weak Topology | Let $\GF \in \set {\R, \C}$.
Let $X$ be a topological vector space over $\GF$ with weak topology $w$.
Let $X^\ast$ be the topological dual space of $X$.
Let $f : X \to \GF$ be a linear functional.
Then $f$ is $w$-continuous {{iff}} $f \in X^\ast$.
That is:
:$\struct {X, w}^\ast = X^\ast$
{{explain|Better to introdu... | This is precisely Continuity of Linear Functionals in Initial Topology on Vector Space Generated by Linear Functionals, taking $F = X^\ast$.
{{qed}}
Category:Weak Topologies on Topological Vector Spaces
n7q6ziw6u8e8gpvaz2rzc2qfjlfaxeq | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$ with [[Definition:Weak Topology on Topological Vector Space|weak topology]] $w$.
Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $X$.
Let $f : X \to \GF$ be a [[... | This is precisely [[Continuity of Linear Functionals in Initial Topology on Vector Space Generated by Linear Functionals]], taking $F = X^\ast$.
{{qed}}
[[Category:Weak Topologies on Topological Vector Spaces]]
n7q6ziw6u8e8gpvaz2rzc2qfjlfaxeq | Characterization of Continuity of Linear Functional in Weak Topology | https://proofwiki.org/wiki/Characterization_of_Continuity_of_Linear_Functional_in_Weak_Topology | https://proofwiki.org/wiki/Characterization_of_Continuity_of_Linear_Functional_in_Weak_Topology | [
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Topological Dual Space",
"Definition:Linear Functional",
"Definition:Continuous Mapping (Topology)"
] | [
"Continuity of Linear Functionals in Weak Topology Induced by Pair of Vector Spaces with Bilinear Mapping/Corollary",
"Category:Weak Topologies on Topological Vector Spaces"
] |
proofwiki-20470 | Equivalence of Definitions of Weak Convergence on Topological Vector Space | Let $\GF \in \set {\R, \C}$.
Let $X$ be a topological vector space over $\GF$.
Let $X^\ast$ be the topological dual space of $X$.
Suppose that:
:for each $x, y \in X$ with $x \ne y$, there exists $f \in X^\ast$ such that $\map f x \ne \map f y$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.
Let $x... | === Definition 1 implies Definition 2 ===
Suppose that $\sequence {x_n}_{n \mathop \in \N}$ converges in $\struct {X, w}$.
Let $f \in X^\ast$ and $\epsilon > 0$.
We want to show that:
:$\map f {x_n} \to \map f x$ in $\GF$.
Let $\map {B_r} {\lambda, \GF}$ be the open ball in $\GF$ of radius $r$ and Centre $\lambda$.
... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $X$.
Suppose that:
:for each $x, y \in X$ with $x \ne y$, there exists $f \in X^\ast$ such that $\map f x \ne ... | === Definition 1 implies Definition 2 ===
Suppose that $\sequence {x_n}_{n \mathop \in \N}$ [[Definition:Convergent Sequence (Topology)|converges]] in $\struct {X, w}$.
Let $f \in X^\ast$ and $\epsilon > 0$.
We want to show that:
:$\map f {x_n} \to \map f x$ in $\GF$.
Let $\map {B_r} {\lambda, \GF}$ be the [[De... | Equivalence of Definitions of Weak Convergence on Topological Vector Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weak_Convergence_on_Topological_Vector_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weak_Convergence_on_Topological_Vector_Space | [
"Weak Convergence (Topological Vector Spaces)",
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Topological Dual Space",
"Definition:Sequence"
] | [
"Definition:Convergent Sequence/Topology",
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Characterization of Continuity of Linear Functional in Weak Topology",
"Definition:Continuous Mapping (Topology)",
"Definition:Initial Topology",
"Definition:Weakly Open Se... |
proofwiki-20471 | Wave Profile of Harmonic Wave | Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.
Then the '''wave profile''' of $\phi$ can be expressed as:
:$\paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$ | By definition, a '''harmonic wave''' is a wave whose wave profile can be expressed as a sine curve.
By definition, a sine curve can be expressed in the form:
:$\map \phi x = a \map \sin {\omega x + \epsilon}$
where $a$, $\omega$ and $\epsilon$ are arbitrary constants.
We select $\epsilon$ so as to set:
:$\epsilon = \df... | Let $\phi$ be a [[Definition:Harmonic Wave|harmonic wave]] which is [[Definition:Direction of Propagation of Wave|propagated]] along the [[Definition:X-Axis|$x$-axis]] in the [[Definition:Positive Direction|positive direction]] with [[Definition:Constant|constant]] [[Definition:Velocity|velocity]] $c$.
Then the '''[[... | By definition, a '''[[Definition:Harmonic Wave|harmonic wave]]''' is a [[Definition:Wave|wave]] whose [[Definition:Wave Profile|wave profile]] can be expressed as a [[Definition:Sine Curve|sine curve]].
By definition, a [[Definition:Sine Curve|sine curve]] can be expressed in the form:
:$\map \phi x = a \map \sin {\om... | Wave Profile of Harmonic Wave | https://proofwiki.org/wiki/Wave_Profile_of_Harmonic_Wave | https://proofwiki.org/wiki/Wave_Profile_of_Harmonic_Wave | [
"Harmonic Waves",
"Wave Profiles"
] | [
"Definition:Harmonic Wave",
"Definition:Wave/Direction of Propagation",
"Definition:Axis/X-Axis",
"Definition:Axis/Positive Direction",
"Definition:Constant",
"Definition:Velocity",
"Definition:Wave Profile"
] | [
"Definition:Harmonic Wave",
"Definition:Wave",
"Definition:Wave Profile",
"Definition:Sine Curve",
"Definition:Sine Curve",
"Definition:Constant",
"Definition:Time",
"Definition:Equation",
"Definition:Wave",
"Definition:Sine Curve",
"Definition:Wave Profile"
] |
proofwiki-20472 | Equation of Harmonic Wave | Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.
Then the '''disturbance''' of $\phi$ at point $x$ and time $t$ can be expressed using the equation:
:$\map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$ | From Wave Profile of Harmonic Wave:
:$\forall x \in \R: \paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$
From Equation of Wave with Constant Velocity:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
Hence the result.
{{qed}} | Let $\phi$ be a [[Definition:Harmonic Wave|harmonic wave]] which is [[Definition:Direction of Propagation of Wave|propagated]] along the [[Definition:X-Axis|$x$-axis]] in the [[Definition:Positive Direction|positive direction]] with [[Definition:Constant|constant]] [[Definition:Velocity|velocity]] $c$.
Then the '''[[D... | From [[Wave Profile of Harmonic Wave]]:
:$\forall x \in \R: \paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$
From [[Equation of Wave with Constant Velocity]]:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
Hence the result.
{{qed}} | Equation of Harmonic Wave | https://proofwiki.org/wiki/Equation_of_Harmonic_Wave | https://proofwiki.org/wiki/Equation_of_Harmonic_Wave | [
"Equation of Harmonic Wave",
"Harmonic Waves"
] | [
"Definition:Harmonic Wave",
"Definition:Wave/Direction of Propagation",
"Definition:Axis/X-Axis",
"Definition:Axis/Positive Direction",
"Definition:Constant",
"Definition:Velocity",
"Definition:Wave/Disturbance",
"Definition:Point",
"Definition:Instant of Time",
"Definition:Equation"
] | [
"Wave Profile of Harmonic Wave",
"Equation of Wave with Constant Velocity"
] |
proofwiki-20473 | Wavelength of Harmonic Wave | Let $\phi$ be a harmonic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''wavelength''' $\lambda$ of $\phi$ can be expressed as:
:$\lambda = \dfrac {2 \pi} \omega$ | By definition, $\lambda$ is the period of the wave profile of $\phi$.
From Wave Profile of Harmonic Wave, the wave profile of $\phi$ is given by:
:$\paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$
From Period of Real Cosine Function:
:$\paren {\map \phi x}_{t \mathop = 0} = a \cos {\omega x + 2 \pi}$
So the peri... | Let $\phi$ be a [[Definition:Harmonic Wave|harmonic wave]] expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''[[Definition:Wavelength of Harmonic Wave|wavelength]]''' $\lambda$ of $\phi$ can be expressed as:
:$\lambda = \dfrac {2 \pi} \omega$ | By definition, $\lambda$ is the [[Definition:Period of Periodic Real Function|period]] of the [[Definition:Wave Profile|wave profile]] of $\phi$.
From [[Wave Profile of Harmonic Wave]], the [[Definition:Wave Profile|wave profile]] of $\phi$ is given by:
:$\paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$
From ... | Wavelength of Harmonic Wave | https://proofwiki.org/wiki/Wavelength_of_Harmonic_Wave | https://proofwiki.org/wiki/Wavelength_of_Harmonic_Wave | [
"Wavelength",
"Harmonic Waves"
] | [
"Definition:Harmonic Wave",
"Definition:Harmonic Wave/Wavelength"
] | [
"Definition:Periodic Real Function/Period",
"Definition:Wave Profile",
"Wave Profile of Harmonic Wave",
"Definition:Wave Profile",
"Period of Real Cosine Function",
"Definition:Periodic Real Function/Period",
"Definition:Harmonic Wave/Wavelength"
] |
proofwiki-20474 | Period of Real Cosine Function | The period of the real cosine function is $2 \pi$.
That is, $2 \pi$ is the smallest value $L \in \R_{>0}$ such that:
:$\forall x \in \R: \cos x = \map \cos {x + L}$ | From Sine and Cosine are Periodic on Reals, we have that $\cos$ is a periodic real function.
Let $L$ be that period.
From Cosine of Angle plus Full Angle:
:$\map \cos {x + 2 \pi} = \cos x$
So $L = 2 \pi$ satisfies:
:$\forall x \in \R: \cos x = \map \cos {x + L}$
It remains to be shown that $2 \pi$ is the smallest such ... | The [[Definition:Period of Periodic Real Function|period]] of the [[Definition:Real Cosine Function|real cosine function]] is $2 \pi$.
That is, $2 \pi$ is the [[Definition:Smallest|smallest]] value $L \in \R_{>0}$ such that:
:$\forall x \in \R: \cos x = \map \cos {x + L}$ | From [[Sine and Cosine are Periodic on Reals]], we have that $\cos$ is a [[Definition:Periodic Real Function|periodic real function]].
Let $L$ be that [[Definition:Period of Periodic Real Function|period]].
From [[Cosine of Angle plus Full Angle]]:
:$\map \cos {x + 2 \pi} = \cos x$
So $L = 2 \pi$ satisfies:
:$\foral... | Period of Real Cosine Function | https://proofwiki.org/wiki/Period_of_Real_Cosine_Function | https://proofwiki.org/wiki/Period_of_Real_Cosine_Function | [
"Cosine Function",
"Periodic Functions"
] | [
"Definition:Periodic Real Function/Period",
"Definition:Cosine/Real Function",
"Definition:Smallest"
] | [
"Sine and Cosine are Periodic on Reals",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Cosine of Angle plus Full Angle",
"Cosine of Right Angle",
"Cosine of Three Right Angles",
"Cosine of Angle plus Full Angle",
"Definition:Periodic Real Function/Period",
"Defini... |
proofwiki-20475 | Period of Real Sine Function | The period of the real sine function is $2 \pi$.
That is, $2 \pi$ is the smallest value $L \in \R_{>0}$ such that:
:$\forall x \in \R: \sin x = \map \sin {x + L}$ | From Sine and Cosine are Periodic on Reals, we have that $\sin$ is a periodic real function.
Let $L$ be that period.
From Sine of Angle plus Full Angle:
:$\map \sin {x + 2 \pi} = \sin x$
So $L = 2 \pi$ satisfies:
:$\forall x \in \R: \sin x = \map \sin {x + L}$
It remains to be shown that $2 \pi$ is the smallest such $L... | The [[Definition:Period of Periodic Real Function|period]] of the [[Definition:Real Sine Function|real sine function]] is $2 \pi$.
That is, $2 \pi$ is the [[Definition:Smallest|smallest]] value $L \in \R_{>0}$ such that:
:$\forall x \in \R: \sin x = \map \sin {x + L}$ | From [[Sine and Cosine are Periodic on Reals]], we have that $\sin$ is a [[Definition:Periodic Real Function|periodic real function]].
Let $L$ be that [[Definition:Period of Periodic Real Function|period]].
From [[Sine of Angle plus Full Angle]]:
:$\map \sin {x + 2 \pi} = \sin x$
So $L = 2 \pi$ satisfies:
:$\forall ... | Period of Real Sine Function | https://proofwiki.org/wiki/Period_of_Real_Sine_Function | https://proofwiki.org/wiki/Period_of_Real_Sine_Function | [
"Sine Function",
"Periodic Functions"
] | [
"Definition:Periodic Real Function/Period",
"Definition:Sine/Real Function",
"Definition:Smallest"
] | [
"Sine and Cosine are Periodic on Reals",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Sine of Angle plus Full Angle",
"Sine of Zero is Zero",
"Sine of Straight Angle",
"Sine of Angle plus Full Angle",
"Definition:Periodic Real Function/Period",
"Definition:Sine/R... |
proofwiki-20476 | Period of Harmonic Wave | Let $\phi$ be a harmonic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''period''' $\tau$ of $\phi$ can be expressed as:
:$\tau = \dfrac \lambda c$
where $\lambda$ is the wavelength of $\phi$. | By definition, a harmonic wave is an instance of a periodic wave.
Hence Period of Periodic Wave can be used directly.
{{qed}} | Let $\phi$ be a [[Definition:Harmonic Wave|harmonic wave]] expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''[[Definition:Period of Harmonic Wave|period]]''' $\tau$ of $\phi$ can be expressed as:
:$\tau = \dfrac \lambda c$
where $\lambda$ is the [[Definition:Wavele... | By definition, a [[Definition:Harmonic Wave|harmonic wave]] is an instance of a [[Definition:Periodic Wave|periodic wave]].
Hence [[Period of Periodic Wave]] can be used directly.
{{qed}} | Period of Harmonic Wave/Proof 1 | https://proofwiki.org/wiki/Period_of_Harmonic_Wave | https://proofwiki.org/wiki/Period_of_Harmonic_Wave/Proof_1 | [
"Period of Harmonic Wave",
"Harmonic Waves"
] | [
"Definition:Harmonic Wave",
"Definition:Harmonic Wave/Period",
"Definition:Harmonic Wave/Wavelength"
] | [
"Definition:Harmonic Wave",
"Definition:Periodic Wave",
"Period of Periodic Wave"
] |
proofwiki-20477 | Period of Harmonic Wave | Let $\phi$ be a harmonic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''period''' $\tau$ of $\phi$ can be expressed as:
:$\tau = \dfrac \lambda c$
where $\lambda$ is the wavelength of $\phi$. | By definition, $\tau$ is the time taken for one complete wavelength of $\phi$ to pass an arbitrary point on the $x$-axis.
From Equation of Harmonic Wave, we have:
:$(1): \quad \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
From Wavelength of Harmonic Wave:
:$\lambda = \dfrac {2 \pi} \omega$
Hence:
:$\omega ... | Let $\phi$ be a [[Definition:Harmonic Wave|harmonic wave]] expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''[[Definition:Period of Harmonic Wave|period]]''' $\tau$ of $\phi$ can be expressed as:
:$\tau = \dfrac \lambda c$
where $\lambda$ is the [[Definition:Wavele... | By definition, $\tau$ is the [[Definition:Time|time]] taken for one complete [[Definition:Wavelength of Harmonic Wave|wavelength]] of $\phi$ to pass an arbitrary [[Definition:Point|point]] on the [[Definition:X-Axis|$x$-axis]].
From [[Equation of Harmonic Wave]], we have:
:$(1): \quad \map \phi {x, t} = a \map \cos {... | Period of Harmonic Wave/Proof 2 | https://proofwiki.org/wiki/Period_of_Harmonic_Wave | https://proofwiki.org/wiki/Period_of_Harmonic_Wave/Proof_2 | [
"Period of Harmonic Wave",
"Harmonic Waves"
] | [
"Definition:Harmonic Wave",
"Definition:Harmonic Wave/Period",
"Definition:Harmonic Wave/Wavelength"
] | [
"Definition:Time",
"Definition:Harmonic Wave/Wavelength",
"Definition:Point",
"Definition:Axis/X-Axis",
"Equation of Harmonic Wave",
"Wavelength of Harmonic Wave"
] |
proofwiki-20478 | Frequency of Harmonic Wave | Let $\phi$ be a harmonic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''frequency''' $\nu$ of $\phi$ can be expressed as:
:$\nu = \dfrac 1 \tau$
where $\tau$ is the period of $\phi$. | By definition, a harmonic wave is an instance of a periodic wave.
Hence Frequency of Periodic Wave can be used directly.
{{qed}} | Let $\phi$ be a [[Definition:Harmonic Wave|harmonic wave]] expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''[[Definition:Frequency of Harmonic Wave|frequency]]''' $\nu$ of $\phi$ can be expressed as:
:$\nu = \dfrac 1 \tau$
where $\tau$ is the [[Definition:Period o... | By definition, a [[Definition:Harmonic Wave|harmonic wave]] is an instance of a [[Definition:Periodic Wave|periodic wave]].
Hence [[Frequency of Periodic Wave]] can be used directly.
{{qed}} | Frequency of Harmonic Wave/Proof 1 | https://proofwiki.org/wiki/Frequency_of_Harmonic_Wave | https://proofwiki.org/wiki/Frequency_of_Harmonic_Wave/Proof_1 | [
"Frequency of Harmonic Wave",
"Frequency",
"Harmonic Waves"
] | [
"Definition:Harmonic Wave",
"Definition:Harmonic Wave/Frequency",
"Definition:Harmonic Wave/Period"
] | [
"Definition:Harmonic Wave",
"Definition:Periodic Wave",
"Frequency of Periodic Wave"
] |
proofwiki-20479 | Velocity of Harmonic Wave is Wavelength times Frequency | Let $\phi$ be a harmonic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
where $c$ is the velocity of $\phi$.
Then:
:$c = \nu \lambda$
where:
:$\nu$ is the frequency of $\phi$
:$\lambda$ is the wavelength of $\phi$. | By definition, a harmonic wave is an instance of a periodic wave.
Hence Velocity of Periodic Wave is Wavelength times Frequency can be used directly. | Let $\phi$ be a [[Definition:Harmonic Wave|harmonic wave]] expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
where $c$ is the [[Definition:Velocity|velocity]] of $\phi$.
Then:
:$c = \nu \lambda$
where:
:$\nu$ is the [[Definition:Frequency of Harmonic Wave|frequency]] of $... | By definition, a [[Definition:Harmonic Wave|harmonic wave]] is an instance of a [[Definition:Periodic Wave|periodic wave]].
Hence [[Velocity of Periodic Wave is Wavelength times Frequency]] can be used directly. | Velocity of Harmonic Wave is Wavelength times Frequency/Proof 1 | https://proofwiki.org/wiki/Velocity_of_Harmonic_Wave_is_Wavelength_times_Frequency | https://proofwiki.org/wiki/Velocity_of_Harmonic_Wave_is_Wavelength_times_Frequency/Proof_1 | [
"Velocity of Harmonic Wave is Wavelength times Frequency",
"Velocity of Periodic Wave is Wavelength times Frequency",
"Harmonic Waves"
] | [
"Definition:Harmonic Wave",
"Definition:Velocity",
"Definition:Harmonic Wave/Frequency",
"Definition:Harmonic Wave/Wavelength"
] | [
"Definition:Harmonic Wave",
"Definition:Periodic Wave",
"Velocity of Periodic Wave is Wavelength times Frequency"
] |
proofwiki-20480 | Equation of Harmonic Wave/Wavelength and Period | :$\map \phi {x, t} = a \map \cos {2 \pi \paren {\dfrac x \lambda - \dfrac t \tau} }$
where:
:$\lambda$ is the wavelength of $\phi$
:$\tau$ is the period of $\phi$. | {{begin-eqn}}
{{eqn | l = \map \phi {x, t}
| r = a \map \cos {\dfrac {2 \pi} \lambda \paren {x - c t} }
| c = Equation of Harmonic Wave in terms of Wavelength and Velocity
}}
{{eqn | r = a \map \cos {\dfrac {2 \pi} \lambda \paren {x - \dfrac \lambda \tau t} }
| c = Period of Harmonic Wave: $\tau = \df... | :$\map \phi {x, t} = a \map \cos {2 \pi \paren {\dfrac x \lambda - \dfrac t \tau} }$
where:
:$\lambda$ is the [[Definition:Wavelength of Harmonic Wave|wavelength]] of $\phi$
:$\tau$ is the [[Definition:Period of Harmonic Wave|period]] of $\phi$. | {{begin-eqn}}
{{eqn | l = \map \phi {x, t}
| r = a \map \cos {\dfrac {2 \pi} \lambda \paren {x - c t} }
| c = [[Equation of Harmonic Wave/Wavelength and Velocity|Equation of Harmonic Wave in terms of Wavelength and Velocity]]
}}
{{eqn | r = a \map \cos {\dfrac {2 \pi} \lambda \paren {x - \dfrac \lambda \tau... | Equation of Harmonic Wave/Wavelength and Period | https://proofwiki.org/wiki/Equation_of_Harmonic_Wave/Wavelength_and_Period | https://proofwiki.org/wiki/Equation_of_Harmonic_Wave/Wavelength_and_Period | [
"Equation of Harmonic Wave"
] | [
"Definition:Harmonic Wave/Wavelength",
"Definition:Harmonic Wave/Period"
] | [
"Equation of Harmonic Wave/Wavelength and Velocity",
"Period of Harmonic Wave"
] |
proofwiki-20481 | Equation of Harmonic Wave/Wavelength and Frequency | :$\map \phi {x, t} = a \map \cos {2 \pi \paren {\dfrac x \lambda - \nu t} }$
where:
:$\lambda$ is the wavelength of $\phi$
:$\nu$ is the frequency of $\phi$. | {{begin-eqn}}
{{eqn | l = \map \phi {x, t}
| r = a \map \cos {2 \pi \paren {\dfrac x \lambda - \dfrac t \tau} }
| c = Equation of Harmonic Wave in terms of Wavelength and Period: $\tau$ is the period of $\phi$
}}
{{eqn | r = a \map \cos {2 \pi \paren {\dfrac x \lambda - \dfrac 1 \tau t} }
| c = rearra... | :$\map \phi {x, t} = a \map \cos {2 \pi \paren {\dfrac x \lambda - \nu t} }$
where:
:$\lambda$ is the [[Definition:Wavelength of Harmonic Wave|wavelength]] of $\phi$
:$\nu$ is the [[Definition:Frequency of Harmonic Wave|frequency]] of $\phi$. | {{begin-eqn}}
{{eqn | l = \map \phi {x, t}
| r = a \map \cos {2 \pi \paren {\dfrac x \lambda - \dfrac t \tau} }
| c = [[Equation of Harmonic Wave/Wavelength and Period|Equation of Harmonic Wave in terms of Wavelength and Period]]: $\tau$ is the [[Definition:Period of Harmonic Wave|period]] of $\phi$
}}
{{eq... | Equation of Harmonic Wave/Wavelength and Frequency | https://proofwiki.org/wiki/Equation_of_Harmonic_Wave/Wavelength_and_Frequency | https://proofwiki.org/wiki/Equation_of_Harmonic_Wave/Wavelength_and_Frequency | [
"Equation of Harmonic Wave"
] | [
"Definition:Harmonic Wave/Wavelength",
"Definition:Harmonic Wave/Frequency"
] | [
"Equation of Harmonic Wave/Wavelength and Period",
"Definition:Harmonic Wave/Period",
"Frequency of Harmonic Wave"
] |
proofwiki-20482 | Equation of Harmonic Wave/Wavelength and Velocity | :$\map \phi {x, t} = a \map \cos {\dfrac {2 \pi} \lambda \paren {x - c t} }$
where $\lambda$ is the wavelength of $\phi$ | {{begin-eqn}}
{{eqn | l = \map \phi {x, t}
| r = a \map \cos {\omega \paren {x - c t} }
| c = Equation of Harmonic Wave
}}
{{eqn | r = a \map \cos {\dfrac {2 \pi} \lambda \paren {x - c t} }
| c = Wavelength of Harmonic Wave: $\lambda = \dfrac {2 \pi} \omega$
}}
{{end-eqn}}
{{qed}} | :$\map \phi {x, t} = a \map \cos {\dfrac {2 \pi} \lambda \paren {x - c t} }$
where $\lambda$ is the [[Definition:Wavelength of Harmonic Wave|wavelength]] of $\phi$ | {{begin-eqn}}
{{eqn | l = \map \phi {x, t}
| r = a \map \cos {\omega \paren {x - c t} }
| c = [[Equation of Harmonic Wave]]
}}
{{eqn | r = a \map \cos {\dfrac {2 \pi} \lambda \paren {x - c t} }
| c = [[Wavelength of Harmonic Wave]]: $\lambda = \dfrac {2 \pi} \omega$
}}
{{end-eqn}}
{{qed}} | Equation of Harmonic Wave/Wavelength and Velocity | https://proofwiki.org/wiki/Equation_of_Harmonic_Wave/Wavelength_and_Velocity | https://proofwiki.org/wiki/Equation_of_Harmonic_Wave/Wavelength_and_Velocity | [
"Equation of Harmonic Wave"
] | [
"Definition:Harmonic Wave/Wavelength"
] | [
"Equation of Harmonic Wave",
"Wavelength of Harmonic Wave"
] |
proofwiki-20483 | Wave Number of Harmonic Wave | Let $\phi$ be a harmonic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''wave number''' $k$ of $\phi$ can be expressed as:
:$k = \dfrac 1 \lambda$
where $\lambda$ is the wavelength of $\phi$. | By definition, a harmonic wave is an instance of a periodic wave.
Hence Wave Number of Periodic Wave can be used directly.
{{qed}} | Let $\phi$ be a [[Definition:Harmonic Wave|harmonic wave]] expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''[[Definition:Wave Number of Harmonic Wave|wave number]]''' $k$ of $\phi$ can be expressed as:
:$k = \dfrac 1 \lambda$
where $\lambda$ is the [[Definition:Wa... | By definition, a [[Definition:Harmonic Wave|harmonic wave]] is an instance of a [[Definition:Periodic Wave|periodic wave]].
Hence [[Wave Number of Periodic Wave]] can be used directly.
{{qed}} | Wave Number of Harmonic Wave/Proof 1 | https://proofwiki.org/wiki/Wave_Number_of_Harmonic_Wave | https://proofwiki.org/wiki/Wave_Number_of_Harmonic_Wave/Proof_1 | [
"Wave Number of Harmonic Wave",
"Wave Number",
"Harmonic Waves"
] | [
"Definition:Harmonic Wave",
"Definition:Harmonic Wave/Wave Number",
"Definition:Harmonic Wave/Wavelength"
] | [
"Definition:Harmonic Wave",
"Definition:Periodic Wave",
"Wave Number of Periodic Wave"
] |
proofwiki-20484 | Wave Number of Harmonic Wave | Let $\phi$ be a harmonic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''wave number''' $k$ of $\phi$ can be expressed as:
:$k = \dfrac 1 \lambda$
where $\lambda$ is the wavelength of $\phi$. | By definition, $k$ is the number of complete wavelengths of $\phi$ per unit displacement along the $x$-axis.
By definition, $\lambda$ is the period of the wave profile of $\phi$.
So between two points at unit distance apart, there are $\dfrac 1 \lambda$ wavelengths of $\phi$.
The result follows by definition of wave nu... | Let $\phi$ be a [[Definition:Harmonic Wave|harmonic wave]] expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$
The '''[[Definition:Wave Number of Harmonic Wave|wave number]]''' $k$ of $\phi$ can be expressed as:
:$k = \dfrac 1 \lambda$
where $\lambda$ is the [[Definition:Wa... | By definition, $k$ is the number of complete [[Definition:Wavelength of Harmonic Wave|wavelengths]] of $\phi$ per unit [[Definition:Displacement|displacement]] along the [[Definition:X-Axis|$x$-axis]].
By definition, $\lambda$ is the [[Definition:Period of Periodic Real Function|period]] of the [[Definition:Wave Profi... | Wave Number of Harmonic Wave/Proof 2 | https://proofwiki.org/wiki/Wave_Number_of_Harmonic_Wave | https://proofwiki.org/wiki/Wave_Number_of_Harmonic_Wave/Proof_2 | [
"Wave Number of Harmonic Wave",
"Wave Number",
"Harmonic Waves"
] | [
"Definition:Harmonic Wave",
"Definition:Harmonic Wave/Wave Number",
"Definition:Harmonic Wave/Wavelength"
] | [
"Definition:Harmonic Wave/Wavelength",
"Definition:Displacement",
"Definition:Axis/X-Axis",
"Definition:Periodic Real Function/Period",
"Definition:Wave Profile",
"Definition:Point",
"Definition:Distance between Points",
"Definition:Harmonic Wave/Wavelength",
"Definition:Harmonic Wave/Wave Number"
] |
proofwiki-20485 | Equation of Harmonic Wave/Wave Number and Frequency | :$\map \phi {x, t} = a \map \cos {2 \pi \paren {k x - \nu t} }$
where:
:$k$ is the wave number of $\phi$
:$\nu$ is the frequency of $\phi$. | {{begin-eqn}}
{{eqn | l = \map \phi {x, t}
| r = a \map \cos {2 \pi \paren {\dfrac x \lambda - \nu t} }
| c = Equation of Harmonic Wave in terms of Wavelength and Frequency: $\lambda$ is the wavelength of $\phi$
}}
{{eqn | r = a \map \cos {2 \pi \paren {\dfrac 1 \lambda x - \nu t} }
| c = rearranging
... | :$\map \phi {x, t} = a \map \cos {2 \pi \paren {k x - \nu t} }$
where:
:$k$ is the [[Definition:Wave Number of Harmonic Wave|wave number]] of $\phi$
:$\nu$ is the [[Definition:Frequency of Harmonic Wave|frequency]] of $\phi$. | {{begin-eqn}}
{{eqn | l = \map \phi {x, t}
| r = a \map \cos {2 \pi \paren {\dfrac x \lambda - \nu t} }
| c = [[Equation of Harmonic Wave/Wavelength and Frequency|Equation of Harmonic Wave in terms of Wavelength and Frequency]]: $\lambda$ is the [[Definition:Wavelength of Harmonic Wave|wavelength]] of $\phi... | Equation of Harmonic Wave/Wave Number and Frequency | https://proofwiki.org/wiki/Equation_of_Harmonic_Wave/Wave_Number_and_Frequency | https://proofwiki.org/wiki/Equation_of_Harmonic_Wave/Wave_Number_and_Frequency | [
"Equation of Harmonic Wave"
] | [
"Definition:Harmonic Wave/Wave Number",
"Definition:Harmonic Wave/Frequency"
] | [
"Equation of Harmonic Wave/Wavelength and Frequency",
"Definition:Harmonic Wave/Wavelength",
"Wave Number of Harmonic Wave"
] |
proofwiki-20486 | Period of Periodic Wave | Let $\phi$ be a periodic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = \map f {x - c t}$
The '''period''' $\tau$ of $\phi$ can be expressed as:
:$\tau = \dfrac \lambda c$
where $\lambda$ is the wavelength of $\phi$. | By definition, $\tau$ is the time taken for one complete wavelength of $\phi$ to pass an arbitrary point on the $x$-axis.
We have:
:$\map \phi {x, t} = \map f {x - c t} = \map f {x - c t + \lambda}$
It follows that $x - c t$ must pass through a complete cycle of values as $t$ is increased by $\tau$.
Thus:
:$\lambda = c... | Let $\phi$ be a [[Definition:Periodic Wave|periodic wave]] expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = \map f {x - c t}$
The '''[[Definition:Period of Periodic Wave|period]]''' $\tau$ of $\phi$ can be expressed as:
:$\tau = \dfrac \lambda c$
where $\lambda$ is the [[Definition:Wavelength of Periodic Wave|... | By definition, $\tau$ is the [[Definition:Time|time]] taken for one complete [[Definition:Wavelength of Periodic Wave|wavelength]] of $\phi$ to pass an arbitrary [[Definition:Point|point]] on the [[Definition:X-Axis|$x$-axis]].
We have:
:$\map \phi {x, t} = \map f {x - c t} = \map f {x - c t + \lambda}$
It follows t... | Period of Periodic Wave | https://proofwiki.org/wiki/Period_of_Periodic_Wave | https://proofwiki.org/wiki/Period_of_Periodic_Wave | [
"Period of Periodic Wave",
"Periodic Waves"
] | [
"Definition:Periodic Wave",
"Definition:Periodic Wave/Period",
"Definition:Periodic Wave/Wavelength"
] | [
"Definition:Time",
"Definition:Periodic Wave/Wavelength",
"Definition:Point",
"Definition:Axis/X-Axis"
] |
proofwiki-20487 | Frequency of Periodic Wave | Let $\phi$ be a periodic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = \map f {x - c t}$
The '''frequency''' $\nu$ of $\phi$ can be expressed as:
:$\nu = \dfrac 1 \tau$
where $\tau$ is the period of $\phi$. | By definition, $\nu$ is the number of complete '''wavelengths''' of $\phi$ to pass an arbitrary point on the $x$-axis in unit time.
Let $x_0$ be that arbitrary point.
By definition, $\tau$ is the time taken for one complete wavelength of $\phi$ to pass $x_0$.
So after unit time, $\dfrac 1 \tau$ wavelengths of $\phi$ pa... | Let $\phi$ be a [[Definition:Periodic Wave|periodic wave]] expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = \map f {x - c t}$
The '''[[Definition:Frequency of Periodic Wave|frequency]]''' $\nu$ of $\phi$ can be expressed as:
:$\nu = \dfrac 1 \tau$
where $\tau$ is the [[Definition:Period of Periodic Wave|period... | By definition, $\nu$ is the number of complete '''[[Definition:Wavelength of Periodic Wave|wavelengths]]''' of $\phi$ to pass an arbitrary [[Definition:Point|point]] on the [[Definition:X-Axis|$x$-axis]] in unit [[Definition:Time|time]].
Let $x_0$ be that arbitrary [[Definition:Point|point]].
By definition, $\tau$ is... | Frequency of Periodic Wave | https://proofwiki.org/wiki/Frequency_of_Periodic_Wave | https://proofwiki.org/wiki/Frequency_of_Periodic_Wave | [
"Frequency of Periodic Wave",
"Frequency",
"Periodic Waves"
] | [
"Definition:Periodic Wave",
"Definition:Periodic Wave/Frequency",
"Definition:Periodic Wave/Period"
] | [
"Definition:Periodic Wave/Wavelength",
"Definition:Point",
"Definition:Axis/X-Axis",
"Definition:Time",
"Definition:Point",
"Definition:Time",
"Definition:Periodic Wave/Wavelength",
"Definition:Time",
"Definition:Periodic Wave/Wavelength"
] |
proofwiki-20488 | Velocity of Periodic Wave is Wavelength times Frequency | Let $\phi$ be a periodic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = \map f {x - c t}$
where $c$ is the velocity of $\phi$.
Then:
:$c = \nu \lambda$
where:
:$\nu$ is the frequency of $\phi$
:$\lambda$ is the wavelength of $\phi$. | {{begin-eqn}}
{{eqn | l = \tau
| r = \dfrac \lambda c
| c = Period of Periodic Wave, where $\tau$ is the period of $\phi$
}}
{{eqn | ll= \leadsto
| l = c
| r = \dfrac 1 \tau \times \lambda
| c = algebra
}}
{{eqn | r = \nu \lambda
| c = Frequency of Periodic Wave
}}
{{end-eqn}}
{{qed}... | Let $\phi$ be a [[Definition:Periodic Wave|periodic wave]] expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = \map f {x - c t}$
where $c$ is the [[Definition:Velocity|velocity]] of $\phi$.
Then:
:$c = \nu \lambda$
where:
:$\nu$ is the [[Definition:Frequency of Periodic Wave|frequency]] of $\phi$
:$\lambda$ is th... | {{begin-eqn}}
{{eqn | l = \tau
| r = \dfrac \lambda c
| c = [[Period of Periodic Wave]], where $\tau$ is the [[Definition:Period of Periodic Wave|period]] of $\phi$
}}
{{eqn | ll= \leadsto
| l = c
| r = \dfrac 1 \tau \times \lambda
| c = algebra
}}
{{eqn | r = \nu \lambda
| c = [[Fre... | Velocity of Periodic Wave is Wavelength times Frequency | https://proofwiki.org/wiki/Velocity_of_Periodic_Wave_is_Wavelength_times_Frequency | https://proofwiki.org/wiki/Velocity_of_Periodic_Wave_is_Wavelength_times_Frequency | [
"Periodic Waves"
] | [
"Definition:Periodic Wave",
"Definition:Velocity",
"Definition:Periodic Wave/Frequency",
"Definition:Periodic Wave/Wavelength"
] | [
"Period of Periodic Wave",
"Definition:Periodic Wave/Period",
"Frequency of Periodic Wave"
] |
proofwiki-20489 | Wave Number of Periodic Wave | Let $\phi$ be a periodic wave expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = \map f {x - c t}$
The '''wave number''' $k$ of $\phi$ can be expressed as:
:$k = \dfrac 1 \lambda$
where $\lambda$ is the wavelength of $\phi$. | By definition, $k$ is the number of complete wavelengths of $\phi$ per unit displacement along the $x$-axis.
By definition, $\lambda$ is the period of the wave profile of $\phi$.
So between two points at unit distance apart, there are $\dfrac 1 \lambda$ wavelengths of $\phi$.
The result follows by definition of wave nu... | Let $\phi$ be a [[Definition:Periodic Wave|periodic wave]] expressed as:
:$\forall x, t \in \R: \map \phi {x, t} = \map f {x - c t}$
The '''[[Definition:Wave Number of Periodic Wave|wave number]]''' $k$ of $\phi$ can be expressed as:
:$k = \dfrac 1 \lambda$
where $\lambda$ is the [[Definition:Wavelength of Periodic W... | By definition, $k$ is the number of complete [[Definition:Wavelength of Periodic Wave|wavelengths]] of $\phi$ per unit [[Definition:Displacement|displacement]] along the [[Definition:X-Axis|$x$-axis]].
By definition, $\lambda$ is the [[Definition:Period of Periodic Real Function|period]] of the [[Definition:Wave Profi... | Wave Number of Periodic Wave | https://proofwiki.org/wiki/Wave_Number_of_Periodic_Wave | https://proofwiki.org/wiki/Wave_Number_of_Periodic_Wave | [
"Wave Number of Periodic Wave",
"Wave Number",
"Periodic Waves"
] | [
"Definition:Periodic Wave",
"Definition:Periodic Wave/Wave Number",
"Definition:Periodic Wave/Wavelength"
] | [
"Definition:Periodic Wave/Wavelength",
"Definition:Displacement",
"Definition:Axis/X-Axis",
"Definition:Periodic Real Function/Period",
"Definition:Wave Profile",
"Definition:Point",
"Definition:Distance between Points",
"Definition:Periodic Wave/Wavelength",
"Definition:Periodic Wave/Wave Number"
] |
proofwiki-20490 | Closed Subspace of Lindelöf Space is Lindelöf Space | Let $T = \struct {S, \tau}$ be a Lindelöf space.
Let $C = \struct {H, \tau_H}$ be a subspace of $T$.
Let $C$ be closed in $T$.
Then $\struct {H, \tau}$ is Lindelöf space.
That is, the property of being Lindelöf is weakly hereditary. | Let $T$ be a Lindelöf space.
Let $C$ be a closed subspace of $T$.
Let $\UU$ be an open cover of $H$.
We have that $H$ is closed in $T$.
It follows by definition of closed that $H \setminus C$ is open in $T$.
So if we add $S \setminus H$ to $\UU$, we see that:
:$\UU \cup \set {S \setminus H}$
is also an open cover of $... | Let $T = \struct {S, \tau}$ be a [[Definition:Lindelöf Space|Lindelöf space]].
Let $C = \struct {H, \tau_H}$ be a [[Definition:Topological Subspace|subspace]] of $T$.
Let $C$ be [[Definition:Closed Set (Topology)|closed]] in $T$.
Then $\struct {H, \tau}$ is [[Definition:Lindelöf Space|Lindelöf space]].
That is, t... | Let $T$ be a [[Definition:Lindelöf Space|Lindelöf space]].
Let $C$ be a [[Definition:Closed Set (Topology)|closed]] [[Definition:Topological Subspace|subspace]] of $T$.
Let $\UU$ be an [[Definition:Open Cover|open cover]] of $H$.
We have that $H$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
It follows by... | Closed Subspace of Lindelöf Space is Lindelöf Space | https://proofwiki.org/wiki/Closed_Subspace_of_Lindelöf_Space_is_Lindelöf_Space | https://proofwiki.org/wiki/Closed_Subspace_of_Lindelöf_Space_is_Lindelöf_Space | [
"Lindelöf Spaces",
"Examples of Weakly Hereditary Properties"
] | [
"Definition:Lindelöf Space",
"Definition:Topological Subspace",
"Definition:Closed Set/Topology",
"Definition:Lindelöf Space",
"Definition:Property",
"Definition:Lindelöf Space",
"Definition:Weakly Hereditary Property"
] | [
"Definition:Lindelöf Space",
"Definition:Closed Set/Topology",
"Definition:Topological Subspace",
"Definition:Open Cover",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Cover",
"Definition:Compact Topological Space",
"Definitio... |
proofwiki-20491 | Bounded Real-Valued Linear Functional is Real Part of Unique Bounded Complex-Valued Linear Functional | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\C$.
Let $\struct {X_\R, \norm {\, \cdot \,}_\R}$ be the realification of $X$ equipped with the restricted norm.
Let $f : X_\R \to \R$ be a bounded linear functional.
Then there exists a unique bounded linear functional $g : X \to \C$ such that:
... | Let $M > 0$ be such that:
:$\cmod {\map f x} \le M \norm x$
for each $x \in X$.
From Linear Functional on Complex Vector Space is Uniquely Determined by Real Part, there exists a unique linear functional $g : X \to \C$ such that:
:$\map f x = \map \Re {\map g x}$
for each $x \in X$, given by:
:$\map g x = \map f x... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\C$.
Let $\struct {X_\R, \norm {\, \cdot \,}_\R}$ be the [[Definition:Realification of Complex Vector Space|realification]] of $X$ equipped with the [[Definition:Restriction of Mapping|restricted]] [[Definition:No... | Let $M > 0$ be such that:
:$\cmod {\map f x} \le M \norm x$
for each $x \in X$.
From [[Linear Functional on Complex Vector Space is Uniquely Determined by Real Part]], there exists a unique [[Definition:Linear Functional|linear functional]] $g : X \to \C$ such that:
:$\map f x = \map \Re {\map g x}$
for each $... | Bounded Real-Valued Linear Functional is Real Part of Unique Bounded Complex-Valued Linear Functional | https://proofwiki.org/wiki/Bounded_Real-Valued_Linear_Functional_is_Real_Part_of_Unique_Bounded_Complex-Valued_Linear_Functional | https://proofwiki.org/wiki/Bounded_Real-Valued_Linear_Functional_is_Real_Part_of_Unique_Bounded_Complex-Valued_Linear_Functional | [
"Bounded Linear Functionals"
] | [
"Definition:Normed Vector Space",
"Definition:Realification of Complex Vector Space",
"Definition:Restriction/Mapping",
"Definition:Norm/Vector Space",
"Definition:Bounded Linear Functional",
"Definition:Bounded Linear Functional"
] | [
"Linear Functional on Complex Vector Space is Uniquely Determined by Real Part",
"Definition:Linear Functional",
"Definition:Bounded Linear Functional",
"Definition:Bounded Linear Functional",
"Category:Bounded Linear Functionals"
] |
proofwiki-20492 | Hahn-Banach Separation Theorem/Normed Vector Space/Complex Case/Open Convex Set and Convex Set | Let $A \subseteq X$ be an open convex set.
Let $B \subseteq X$ be a convex set disjoint from $A$.
Then there exists $f \in X^\ast$ and $c \in \R$ such that:
:$A \subseteq \set {x \in X : \map \Re {\map f x} < c}$
and:
:$B \subseteq \set {x \in X : \map \Re {\map f x} \ge c}$
That is:
:there exists $f \in X^\ast$ and... | Let $\struct {X_\R, \norm {\, \cdot \,}_\R}$ be the realification of $X$ equipped with the restricted norm.
Applying Hahn-Banach Separation Theorem: Real Case: Open Convex Set and Convex Set to $\struct {X_\R, \norm {\, \cdot \,}_\R}$, there exists a bounded linear functional $g : X_\R \to \R$ and $c \in \R$ such tha... | Let $A \subseteq X$ be an [[Definition:Open Set in Normed Vector Space|open]] [[Definition:Convex Set (Vector Space)|convex set]].
Let $B \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex set]] [[Definition:Disjoint Sets|disjoint]] from $A$.
Then there exists $f \in X^\ast$ and $c \in \R$ such that:
... | Let $\struct {X_\R, \norm {\, \cdot \,}_\R}$ be the [[Definition:Realification of Complex Vector Space|realification]] of $X$ equipped with the [[Definition:Restriction of Mapping|restricted]] [[Definition:Norm on Vector Space|norm]].
Applying [[Hahn-Banach Separation Theorem/Real Case/Open Convex Set and Convex Set... | Hahn-Banach Separation Theorem/Normed Vector Space/Complex Case/Open Convex Set and Convex Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Normed_Vector_Space/Complex_Case/Open_Convex_Set_and_Convex_Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Normed_Vector_Space/Complex_Case/Open_Convex_Set_and_Convex_Set | [
"Hahn-Banach Separation Theorem"
] | [
"Definition:Open Set/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Disjoint Sets"
] | [
"Definition:Realification of Complex Vector Space",
"Definition:Restriction/Mapping",
"Definition:Norm/Vector Space",
"Hahn-Banach Separation Theorem/Normed Vector Space/Real Case/Open Convex Set and Convex Set",
"Definition:Bounded Linear Functional",
"Bounded Real-Valued Linear Functional is Real Part o... |
proofwiki-20493 | Hahn-Banach Separation Theorem/Normed Vector Space/Complex Case/Compact Convex Set and Closed Convex Set | Let $A$ be a compact convex set.
Let $B$ be a closed convex set disjoint from $A$.
Then there exists $f \in X^\ast$, $c \in \R$ and $\epsilon > 0$ such that:
:$A \subseteq \set {x \in X : \map \Re {\map f x} \le c - \epsilon}$
and:
:$B \subseteq \set {x \in X : \map \Re {\map f x} \ge c + \epsilon}$
That is:
:there e... | Let $\struct {X_\R, \norm {\, \cdot \,}_\R}$ be the realification of $X$ equipped with the restricted norm.
Applying Hahn-Banach Separation Theorem: Real Case: Open Convex Set and Convex Set to $\struct {X_\R, \norm {\, \cdot \,}_\R}$, there exists a bounded linear functional $g : X_\R \to \R$, $c \in \R$ and $\epsil... | Let $A$ be a [[Definition:Compact Subset of Normed Vector Space|compact]] [[Definition:Convex Set (Vector Space)|convex set]].
Let $B$ be a [[Definition:Closed Set in Normed Vector Space|closed]] [[Definition:Convex Set (Vector Space)|convex set]] [[Definition:Disjoint Sets|disjoint]] from $A$.
Then there exists $f ... | Let $\struct {X_\R, \norm {\, \cdot \,}_\R}$ be the [[Definition:Realification of Complex Vector Space|realification]] of $X$ equipped with the [[Definition:Restriction of Mapping|restricted]] [[Definition:Norm on Vector Space|norm]].
Applying [[Hahn-Banach Separation Theorem/Real Case/Open Convex Set and Convex Set... | Hahn-Banach Separation Theorem/Normed Vector Space/Complex Case/Compact Convex Set and Closed Convex Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Normed_Vector_Space/Complex_Case/Compact_Convex_Set_and_Closed_Convex_Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Normed_Vector_Space/Complex_Case/Compact_Convex_Set_and_Closed_Convex_Set | [
"Hahn-Banach Separation Theorem"
] | [
"Definition:Compact Space/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Closed Set/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Disjoint Sets"
] | [
"Definition:Realification of Complex Vector Space",
"Definition:Restriction/Mapping",
"Definition:Norm/Vector Space",
"Hahn-Banach Separation Theorem/Normed Vector Space/Real Case/Open Convex Set and Convex Set",
"Definition:Bounded Linear Functional",
"Bounded Real-Valued Linear Functional is Real Part o... |
proofwiki-20494 | Sum of Consecutive Odd Index Fibonacci Numbers | {{begin-eqn}}
{{eqn | l = F_{2 k - 1} + F_{2 k + 1}
| r = \phi^{2 k} + \phi^{-2 k}
| c =
}}
{{end-eqn}}
where:
:$F_k$ denotes the $k$th Fibonacci number
:$\phi$ denotes the golden mean. | {{begin-eqn}}
{{eqn | l = F_{2 k - 1} + F_{2 k + 1}
| r = \paren {\phi^{2 k} - F_{2 k} \phi } + \paren {\phi^{2 k + 2} - F_{2 k + 2} \phi }
| c = Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less
}}
{{eqn | r = \phi^{2 k} + F_{-2 k} \phi + \paren {\phi^{2 k + 2} - F_{2 k + 2} \phi }
... | {{begin-eqn}}
{{eqn | l = F_{2 k - 1} + F_{2 k + 1}
| r = \phi^{2 k} + \phi^{-2 k}
| c =
}}
{{end-eqn}}
where:
:$F_k$ denotes the $k$th [[Definition:Fibonacci Number|Fibonacci number]]
:$\phi$ denotes the [[Definition:Golden Mean|golden mean]]. | {{begin-eqn}}
{{eqn | l = F_{2 k - 1} + F_{2 k + 1}
| r = \paren {\phi^{2 k} - F_{2 k} \phi } + \paren {\phi^{2 k + 2} - F_{2 k + 2} \phi }
| c = [[Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less]]
}}
{{eqn | r = \phi^{2 k} + F_{-2 k} \phi + \paren {\phi^{2 k + 2} - F_{2 k + 2} \phi ... | Sum of Consecutive Odd Index Fibonacci Numbers | https://proofwiki.org/wiki/Sum_of_Consecutive_Odd_Index_Fibonacci_Numbers | https://proofwiki.org/wiki/Sum_of_Consecutive_Odd_Index_Fibonacci_Numbers | [
"Fibonacci Numbers",
"Golden Mean"
] | [
"Definition:Fibonacci Number",
"Definition:Golden Mean"
] | [
"Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less",
"Fibonacci Number with Negative Index",
"Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less",
"Fibonacci Number with Negative Index",
"Fibonacci Number by Golden Mean plus Fibonacci Number of Index One Less",
"... |
proofwiki-20495 | Continuous Linear Transformation Algebra has Two-Sided Identity | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space.
Let $\struct {\map {CL} X, *}$ be an associative algebra.
Then there exists an identity element $I \in \map {CL} X$ such that:
:$\forall x \in X : \map I x = x$ | {{ProofWanted}}
{{tidy|the usual}}
Let $X$ be normed vector space over $K$.
Let $\map \LL {X, X}$ denote the set of all linear transformations from $X$ to itself.
Let $\map C {X, X}$ denote the continuous mapping space from $X$ to itself.
Suppose:
:$\map {CL} X := \map {CL} {X, X}$
Since $\struct {\map {CL} X, *}$ is a... | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\map {CL} X := \map {CL} {X, X}$ be a [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $\struct {\map {CL} X, *}$ be an [[Definition:Associative Algebra|associati... | {{ProofWanted}}
{{tidy|the usual}}
Let $X$ be [[Definition:Normed Vector Space|normed vector space]] over $K$.
Let $\map \LL {X, X}$ denote the [[Definition:Set of All Linear Transformations|set of all linear transformations]] from $X$ to itself.
Let $\map C {X, X}$ denote the [[Definition:Continuous Mapping Space... | Continuous Linear Transformation Algebra has Two-Sided Identity | https://proofwiki.org/wiki/Continuous_Linear_Transformation_Algebra_has_Two-Sided_Identity | https://proofwiki.org/wiki/Continuous_Linear_Transformation_Algebra_has_Two-Sided_Identity | [] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Associative Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Normed Vector Space",
"Definition:Set of All Linear Transformations",
"Definition:Continuous Mapping Space",
"Definition:Associative Algebra",
"Identity Mapping is Continuous",
"Identity Mapping on Normed Vector Space is Bounded Linear Operator",
"Definition:Identity (Abstract Algebra)/Two-S... |
proofwiki-20496 | Continuous Linear Transformation Algebra with Supremum Operator Norm is Normed Algebra | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space.
Let $\struct {\map {CL} X, *}$ be an associative algebra.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Then $\struct {\struct {\map {CL} X, *}, \norm {\, \cdo... | We need to show:
$\norm {a \ast b} \le \norm a \norm b$
{{questionable|Is this really true for any associative bilinear $\ast$? }}
{{ProofWanted}} | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\map {CL} X := \map {CL} {X, X}$ be a [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $\struct {\map {CL} X, *}$ be an [[Definition:Associative Algebra|associati... | We need to show:
$\norm {a \ast b} \le \norm a \norm b$
{{questionable|Is this really true for any associative bilinear $\ast$? }}
{{ProofWanted}} | Continuous Linear Transformation Algebra with Supremum Operator Norm is Normed Algebra | https://proofwiki.org/wiki/Continuous_Linear_Transformation_Algebra_with_Supremum_Operator_Norm_is_Normed_Algebra | https://proofwiki.org/wiki/Continuous_Linear_Transformation_Algebra_with_Supremum_Operator_Norm_is_Normed_Algebra | [] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Associative Algebra",
"Definition:Supremum Operator Norm",
"Definition:Normed Algebra"
] | [] |
proofwiki-20497 | Equation of Wavefront of Plane Wave/Direction Cosine Form | Let $\phi$ be a plane wave.
Let an arbitrary wavefront of $\phi$ be denoted $P$.
Let the direction of propagation of $\phi$ be expressed as:
:$x : y : z = l : m : n$
where $l$, $m$ and $n$ are the direction cosines of the normal to $P$.
{{explain|The notation needs to be explained. This will be done on the Definition:D... | {{ProofWanted|More background needed on $3$D analytic / coordinate geometry. I need to crack on with that.}} | Let $\phi$ be a [[Definition:Plane Wave|plane wave]].
Let an arbitrary [[Definition:Wavefront of Plane Wave|wavefront]] of $\phi$ be denoted $P$.
Let the [[Definition:Direction of Propagation of Wave|direction of propagation]] of $\phi$ be expressed as:
:$x : y : z = l : m : n$
where $l$, $m$ and $n$ are the [[Defini... | {{ProofWanted|More background needed on $3$D analytic / coordinate geometry. I need to crack on with that.}} | Equation of Wavefront of Plane Wave/Direction Cosine Form | https://proofwiki.org/wiki/Equation_of_Wavefront_of_Plane_Wave/Direction_Cosine_Form | https://proofwiki.org/wiki/Equation_of_Wavefront_of_Plane_Wave/Direction_Cosine_Form | [
"Equation of Wavefront of Plane Wave"
] | [
"Definition:Plane Wave",
"Definition:Plane Wave/Wavefront",
"Definition:Wave/Direction of Propagation",
"Definition:Direction Cosines",
"Definition:Normal Vector",
"Definition:Direction Cosines",
"Definition:Constant",
"Definition:Plane Wave"
] | [] |
proofwiki-20498 | Equation of Plane Wave/Direction Cosine Form | Let $\phi$ be a plane wave propagated with velocity $c$.
Let the direction of propagation of $\phi$ be expressed as:
:$x : y : z = l : m : n$
where $l$, $m$ and $n$ are the '''direction cosines''' of the normal to $P$.
Then $\phi$ can be expressed as:
:$\map \phi {x, y, z, t} = \map f {l x + m y + n z - c t}$ | By Equation of Wavefront of Plane Wave, the equation of the wavefront of $\phi$ is given by:
:$l x + m y + n z = K$
Hence it is clear that:
:$\map \phi {x, y, z, t} = \map f {l x + m y + n z - c t}$
is a function which fulfils all the requirements to be a plane wave.
{{handwaving|"It is clear that"}}
Hence $\phi$ as de... | Let $\phi$ be a [[Definition:Plane Wave|plane wave]] [[Definition:Direction of Propagation of Wave|propagated]] with [[Definition:Velocity|velocity]] $c$.
Let the [[Definition:Direction of Propagation of Wave|direction of propagation]] of $\phi$ be expressed as:
:$x : y : z = l : m : n$
where $l$, $m$ and $n$ are the ... | By [[Equation of Wavefront of Plane Wave]], the equation of the [[Definition:Wavefront of Plane Wave|wavefront]] of $\phi$ is given by:
:$l x + m y + n z = K$
Hence it is clear that:
:$\map \phi {x, y, z, t} = \map f {l x + m y + n z - c t}$
is a [[Definition:Real-Valued Function|function]] which fulfils all the requ... | Equation of Plane Wave/Direction Cosine Form | https://proofwiki.org/wiki/Equation_of_Plane_Wave/Direction_Cosine_Form | https://proofwiki.org/wiki/Equation_of_Plane_Wave/Direction_Cosine_Form | [
"Equation of Plane Wave"
] | [
"Definition:Plane Wave",
"Definition:Wave/Direction of Propagation",
"Definition:Velocity",
"Definition:Wave/Direction of Propagation",
"Definition:Direction Cosines",
"Definition:Normal Vector"
] | [
"Equation of Wavefront of Plane Wave",
"Definition:Plane Wave/Wavefront",
"Definition:Real-Valued Function",
"Definition:Plane Wave",
"Definition:Plane Wave",
"Definition:Wave/Direction of Propagation",
"Definition:Velocity",
"Definition:Wave/Direction of Propagation"
] |
proofwiki-20499 | T3 Lindelöf Space is T4 | Let $T = \struct {S, \tau}$ be a $T_3$ Lindelöf topological space.
Then $T$ is a $T_4$ space. | Let $A$ and $B$ be disjoint closed subsets of $T$.
Let $\UU = \set {U \in \tau : U^- \cap B = \O}$.
From Characterization of T3 Space:
:$\forall a \in A : \exists U_a \in \tau: a \in U_a : U_a^- \cap B = \O$
By definition of open cover:
:$\UU$ is an open cover of $A$
From Closed Subspace of Lindelöf Space is Lindelöf S... | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$]] [[Definition:Lindelöf Space|Lindelöf]] [[Definition:Topological Space|topological space]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | Let $A$ and $B$ be [[Definition:Disjoint Sets|disjoint]] [[Definition:Closed Subset|closed]] [[Definition:Subset|subsets]] of $T$.
Let $\UU = \set {U \in \tau : U^- \cap B = \O}$.
From [[Characterization of T3 Space]]:
:$\forall a \in A : \exists U_a \in \tau: a \in U_a : U_a^- \cap B = \O$
By definition of [[Defin... | T3 Lindelöf Space is T4/Proof 1 | https://proofwiki.org/wiki/T3_Lindelöf_Space_is_T4 | https://proofwiki.org/wiki/T3_Lindelöf_Space_is_T4/Proof_1 | [
"T3 Spaces",
"Lindelöf Spaces",
"T4 Spaces",
"T3 Lindelöf Space is T4"
] | [
"Definition:T3 Space",
"Definition:Lindelöf Space",
"Definition:Topological Space",
"Definition:T4 Space"
] | [
"Definition:Disjoint Sets",
"Definition:Closed Subset",
"Definition:Subset",
"Characterization of T3 Space",
"Definition:Open Cover",
"Definition:Open Cover",
"Closed Subspace of Lindelöf Space is Lindelöf Space",
"Definition:Lindelöf Space",
"Definition:Topological Subspace",
"Definition:Topologi... |
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