id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-20900 | Convergent Sequence in Topological Vector Space is Cauchy | Let $\struct {X, \tau}$ be a topological vector space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a convergent sequence with $x_n \to x$.
Then $\sequence {x_n}_{n \mathop \in \N}$ is Cauchy. | Let $V$ be an open neighborhood of ${\mathbf 0}_X$.
From {{Corollary|Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods|1}}, there exists a symmetric open neighborhood $U$ of ${\mathbf 0}_X$ such that:
:$U + U \subseteq V$
Since $x_n \to x$, there exists $N \in \N$ such that:
:... | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Convergent Sequence|convergent sequence]] with $x_n \to x$.
Then $\sequence {x_n}_{n \mathop \in \N}$ is [[Definition:Cauchy Sequence in Topological Vector Space|C... | Let $V$ be an [[Definition:Open Neighborhood|open neighborhood]] of ${\mathbf 0}_X$.
From {{Corollary|Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods|1}}, there exists a [[Definition:Symmetric Subset of Vector Space|symmetric]] [[Definition:Open Neighborhood|open neighborhood... | Convergent Sequence in Topological Vector Space is Cauchy | https://proofwiki.org/wiki/Convergent_Sequence_in_Topological_Vector_Space_is_Cauchy | https://proofwiki.org/wiki/Convergent_Sequence_in_Topological_Vector_Space_is_Cauchy | [
"Topological Vector Spaces",
"Cauchy Sequences in Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Convergent Sequence",
"Definition:Cauchy Sequence/Topological Vector Space"
] | [
"Definition:Open Neighborhood",
"Definition:Symmetric Set/Vector Space",
"Definition:Open Neighborhood",
"Definition:Symmetric Set/Vector Space",
"Definition:Cauchy Sequence/Topological Vector Space",
"Category:Topological Vector Spaces",
"Category:Cauchy Sequences in Topological Vector Spaces"
] |
proofwiki-20901 | Image of Cauchy Sequence in Topological Vector Space is von Neumann-Bounded | Let $\struct {X, \tau}$ be a topological vector space.
Let $\sequence {x_n}_{n \in \N}$ be a Cauchy sequence.
Let:
:$E = \set {x_n : n \in \N}$
Then $E$ is von Neumann-bounded. | Let $W$ be an open neighborhood of ${\mathbf 0}_X$.
From Open Neighborhood of Origin in Topological Vector Space contains Balanced Open Neighborhood, take a balanced open neighborhood of ${\mathbf 0_X}$, $V \subseteq W$.
From {{Corollary|Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neigh... | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]].
Let $\sequence {x_n}_{n \in \N}$ be a [[Definition:Cauchy Sequence in Topological Vector Space|Cauchy sequence]].
Let:
:$E = \set {x_n : n \in \N}$
Then $E$ is [[Definition:Von Neumann-Bounded Subset of Topological Vec... | Let $W$ be an [[Definition:Open Neighborhood|open neighborhood]] of ${\mathbf 0}_X$.
From [[Open Neighborhood of Origin in Topological Vector Space contains Balanced Open Neighborhood]], take a [[Definition:Balanced Set|balanced]] [[Definition:Open Neighborhood|open neighborhood]] of ${\mathbf 0_X}$, $V \subseteq W$. ... | Image of Cauchy Sequence in Topological Vector Space is von Neumann-Bounded | https://proofwiki.org/wiki/Image_of_Cauchy_Sequence_in_Topological_Vector_Space_is_von_Neumann-Bounded | https://proofwiki.org/wiki/Image_of_Cauchy_Sequence_in_Topological_Vector_Space_is_von_Neumann-Bounded | [
"Cauchy Sequences in Topological Vector Spaces",
"Von Neumann-Bounded Subsets of Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Cauchy Sequence/Topological Vector Space",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Definition:Open Neighborhood",
"Open Neighborhood of Origin in Topological Vector Space contains Balanced Open Neighborhood",
"Definition:Balanced Set",
"Definition:Open Neighborhood",
"Definition:Symmetric Set/Vector Space",
"Definition:Open Neighborhood",
"Open Neighborhood of Origin in Topological V... |
proofwiki-20902 | Image of Convergent Sequence in Topological Vector Space is von Neumann-Bounded | Let $\struct {X, \tau}$ be a topological vector space.
Let $\sequence {x_n}_{n \in \N}$ be a convergent sequence with $x_n \to x$.
Let:
:$E = \set {x_n : n \in \N}$
Then $E$ is von Neumann-bounded. | From Convergent Sequence in Topological Vector Space is Cauchy, $\sequence {x_n}_{n \in \N}$ is a Cauchy sequence.
From Image of Cauchy Sequence in Topological Vector Space is von Neumann-Bounded, $E$ is von Neumann-bounded.
{{qed}} | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]].
Let $\sequence {x_n}_{n \in \N}$ be a [[Definition:Convergent Sequence|convergent sequence]] with $x_n \to x$.
Let:
:$E = \set {x_n : n \in \N}$
Then $E$ is [[Definition:Von Neumann-Bounded Subset of Topological Vecto... | From [[Convergent Sequence in Topological Vector Space is Cauchy]], $\sequence {x_n}_{n \in \N}$ is a [[Definition:Cauchy Sequence in Topological Vector Space|Cauchy sequence]].
From [[Image of Cauchy Sequence in Topological Vector Space is von Neumann-Bounded]], $E$ is [[Definition:von Neumann-Bounded Subset of Topol... | Image of Convergent Sequence in Topological Vector Space is von Neumann-Bounded/Proof 1 | https://proofwiki.org/wiki/Image_of_Convergent_Sequence_in_Topological_Vector_Space_is_von_Neumann-Bounded | https://proofwiki.org/wiki/Image_of_Convergent_Sequence_in_Topological_Vector_Space_is_von_Neumann-Bounded/Proof_1 | [
"von Neumann-Bounded Subsets of Topological Vector Spaces",
"Image of Convergent Sequence in Topological Vector Space is von Neumann-Bounded"
] | [
"Definition:Topological Vector Space",
"Definition:Convergent Sequence",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Convergent Sequence in Topological Vector Space is Cauchy",
"Definition:Cauchy Sequence/Topological Vector Space",
"Image of Cauchy Sequence in Topological Vector Space is von Neumann-Bounded",
"Definition:von Neumann-Bounded Subset of Topological Vector Space"
] |
proofwiki-20903 | Image of Convergent Sequence in Topological Vector Space is von Neumann-Bounded | Let $\struct {X, \tau}$ be a topological vector space.
Let $\sequence {x_n}_{n \in \N}$ be a convergent sequence with $x_n \to x$.
Let:
:$E = \set {x_n : n \in \N}$
Then $E$ is von Neumann-bounded. | From Union of Image of Convergent Sequence and Limit in Topological Space is Compact, $E \cup \set x$ is compact.
From Compact Subspace of Topological Vector Space is von Neumann-Bounded, $E \cup \set x$ is von Neumann bounded.
From Subset of von Neumann-Bounded Set is von Neumann-Bounded, $E$ is von Neumann bounded.
{... | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]].
Let $\sequence {x_n}_{n \in \N}$ be a [[Definition:Convergent Sequence|convergent sequence]] with $x_n \to x$.
Let:
:$E = \set {x_n : n \in \N}$
Then $E$ is [[Definition:Von Neumann-Bounded Subset of Topological Vecto... | From [[Union of Image of Convergent Sequence and Limit in Topological Space is Compact]], $E \cup \set x$ is [[Definition:Compact Topological Space|compact]].
From [[Compact Subspace of Topological Vector Space is von Neumann-Bounded]], $E \cup \set x$ is [[Definition:von Neumann-Bounded Subset of Topological Vector S... | Image of Convergent Sequence in Topological Vector Space is von Neumann-Bounded/Proof 2 | https://proofwiki.org/wiki/Image_of_Convergent_Sequence_in_Topological_Vector_Space_is_von_Neumann-Bounded | https://proofwiki.org/wiki/Image_of_Convergent_Sequence_in_Topological_Vector_Space_is_von_Neumann-Bounded/Proof_2 | [
"von Neumann-Bounded Subsets of Topological Vector Spaces",
"Image of Convergent Sequence in Topological Vector Space is von Neumann-Bounded"
] | [
"Definition:Topological Vector Space",
"Definition:Convergent Sequence",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Union of Image of Convergent Sequence and Limit in Topological Space is Compact",
"Definition:Compact Topological Space",
"Compact Subspace of Topological Vector Space is von Neumann-Bounded",
"Definition:von Neumann-Bounded Subset of Topological Vector Space",
"Subset of von Neumann-Bounded Set is von Neu... |
proofwiki-20904 | Von Neumann-Bounded Set in Weak-* Topology of Normed Dual of Banach Space is Norm Bounded | Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,} }$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.
Let $E$ be a von Neumann-bounded subset of $\struct {X^\ast, w^\ast}$.
Then $E$ is a v... | Let $x \in X$.
Then from Open Sets in Weak-* Topology of Topological Vector Space:
:$V = \set {f \in X^\ast : \cmod {\map f x} < 1}$ is an open neighborhood of $\mathbf 0_{X^\ast}$ in $\struct {X^\ast, w^\ast}$.
Then there exists $r > 0$ such that:
:$E \subseteq r V$
Then for each $f \in E$, we have $f = r g$ for so... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]].
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,} }$.
Let $w^\ast$ be the [[Definition:Weak-* Topology|weak-$\ast$ topology]] on $X^... | Let $x \in X$.
Then from [[Open Sets in Weak-* Topology of Topological Vector Space]]:
:$V = \set {f \in X^\ast : \cmod {\map f x} < 1}$ is an [[Definition:Open Neighborhood|open neighborhood]] of $\mathbf 0_{X^\ast}$ in $\struct {X^\ast, w^\ast}$.
Then there exists $r > 0$ such that:
:$E \subseteq r V$
Then fo... | Von Neumann-Bounded Set in Weak-* Topology of Normed Dual of Banach Space is Norm Bounded | https://proofwiki.org/wiki/Von_Neumann-Bounded_Set_in_Weak-*_Topology_of_Normed_Dual_of_Banach_Space_is_Norm_Bounded | https://proofwiki.org/wiki/Von_Neumann-Bounded_Set_in_Weak-*_Topology_of_Normed_Dual_of_Banach_Space_is_Norm_Bounded | [
"Weak-* Topologies",
"Von Neumann-Bounded Subsets of Topological Vector Spaces"
] | [
"Definition:Banach Space",
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Open Sets in Weak-* Topology of Topological Vector Space",
"Definition:Open Neighborhood",
"Definition:Banach Space",
"Banach-Steinhaus Theorem",
"Characterization of von Neumann-Boundedness in Normed Vector Space",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space",
"Category:Weak-* T... |
proofwiki-20905 | Point Evaluations are Continuous Linear Functionals on Space of Complex-Valued Continuous Functions on Compact Hausdorff Space | Let $K$ be a compact Hausdorff space.
Let $\map \CC K$ be the space of complex-valued continuous functions of compact Hausdorff space.
For $k \in K$, define $\delta_k : \map \CC K \to \C$ by:
:$\map {\delta_k} f = \map f k$
for each $f \in \map \CC K$.
Then $\delta_k$ is a continuous linear functional on $\map \CC K... | Let $k \in K$.
$\delta_k$ is clearly linear, so we need to show that it is continuous.
From Continuity of Linear Functionals, it is enough to show that $\delta_k$ is bounded.
For each $f \in \map \CC K$ we have:
:$\ds \cmod {\map f k} \le \sup_{x \in K} \cmod {\map f x}$
That is:
:$\cmod {\map {\delta_k} f} \le \no... | Let $K$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Hausdorff Space|Hausdorff space]].
Let $\map \CC K$ be the [[Definition:Space of Continuous Functions on Compact Hausdorff Space|space of complex-valued continuous functions of compact Hausdorff space]].
For $k \in K$, define $\delta_k : \map... | Let $k \in K$.
$\delta_k$ is clearly [[Definition:Linear Functional|linear]], so we need to show that it is [[Definition:Continuous Mapping|continuous]].
From [[Continuity of Linear Functionals]], it is enough to show that $\delta_k$ is [[Definition:Bounded Linear Functional|bounded]].
For each $f \in \map \CC K$ w... | Point Evaluations are Continuous Linear Functionals on Space of Complex-Valued Continuous Functions on Compact Hausdorff Space | https://proofwiki.org/wiki/Point_Evaluations_are_Continuous_Linear_Functionals_on_Space_of_Complex-Valued_Continuous_Functions_on_Compact_Hausdorff_Space | https://proofwiki.org/wiki/Point_Evaluations_are_Continuous_Linear_Functionals_on_Space_of_Complex-Valued_Continuous_Functions_on_Compact_Hausdorff_Space | [
"Space of Continuous Functions on Compact Hausdorff Space"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Continuous Mapping",
"Definition:Linear Functional"
] | [
"Definition:Linear Functional",
"Definition:Continuous Mapping",
"Continuity of Linear Functionals",
"Definition:Bounded Linear Functional",
"Definition:Singleton",
"Definition:Singleton",
"Urysohn's Lemma",
"Definition:Continuous Function",
"Category:Space of Continuous Functions on Compact Hausdor... |
proofwiki-20906 | Space of Complex-Valued Continuous Functions on Compact Hausdorff Space is Separable iff Space is Metrizable | Let $K$ be a compact Hausdorff space.
Let $\map \CC K$ be the space of complex-valued continuous functions of compact Hausdorff space.
Then $\map \CC K$ is separable {{iff}} the topology on $K$ is metrizable. | === Necessary Condition ===
Suppose that $\map \CC K$ is separable.
Then by Closed Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable, $\struct {B^-_{\map \CC K^\ast}, w^\ast}$ is metrizable.
For each $k \in K$, define $\delta_k : \map \CC K \to \C$ by:
:$\map {\delta_k} f = \map f k$
for... | Let $K$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Hausdorff Space|Hausdorff space]].
Let $\map \CC K$ be the [[Definition:Space of Continuous Functions on Compact Hausdorff Space|space of complex-valued continuous functions of compact Hausdorff space]].
Then $\map \CC K$ is [[Definition:Sep... | === Necessary Condition ===
Suppose that $\map \CC K$ is [[Definition:Separable Space|separable]].
Then by [[Closed Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable]], $\struct {B^-_{\map \CC K^\ast}, w^\ast}$ is [[Definition:Metrizable Space|metrizable]].
For each $k \in K$, define $\... | Space of Complex-Valued Continuous Functions on Compact Hausdorff Space is Separable iff Space is Metrizable | https://proofwiki.org/wiki/Space_of_Complex-Valued_Continuous_Functions_on_Compact_Hausdorff_Space_is_Separable_iff_Space_is_Metrizable | https://proofwiki.org/wiki/Space_of_Complex-Valued_Continuous_Functions_on_Compact_Hausdorff_Space_is_Separable_iff_Space_is_Metrizable | [
"Space of Continuous Functions on Compact Hausdorff Space",
"Metrizable Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Separable Space",
"Definition:Topology",
"Definition:Metrizable Space"
] | [
"Definition:Separable Space",
"Closed Ball in Normed Dual Space of Separable Normed Vector Space is Weak-* Metrizable",
"Definition:Metrizable Space",
"Point Evaluations are Continuous Linear Functionals on Space of Complex-Valued Continuous Functions on Compact Hausdorff Space",
"Definition:Continuous Mapp... |
proofwiki-20907 | Characterization of Paracompactness in T3 Space/Lemma 16 | :$\forall n \in \N_{> 0}, \forall y, z \in X : y \ne z \leadsto \map {A_n} z \cap V_{n+1} \sqbrk {\map {A_n} y} = \O$ | Let $n \in \N_{> 0}$.
Let $y, z \in X : y \ne z$. | :$\forall n \in \N_{> 0}, \forall y, z \in X : y \ne z \leadsto \map {A_n} z \cap V_{n+1} \sqbrk {\map {A_n} y} = \O$ | Let $n \in \N_{> 0}$.
Let $y, z \in X : y \ne z$. | Characterization of Paracompactness in T3 Space/Lemma 16 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_16 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_16 | [
"Characterization of Paracompactness in T3 Space"
] | [] | [] |
proofwiki-20908 | Weak-* Metrizability of Closed Unit Ball in Normed Dual of Normed Vector Space implies Original Space is Separable | Let $X$ be a normed vector space such that:
:$\struct {B_{X^\ast}^-, w^\ast}$ is metrizable
where $X^\ast$ is the normed dual of $X$ and $B_{X^\ast}^-$ is the closed unit ball of $X^\ast$.
Then $X$ is separable. | Let:
:$K = \struct {B_{X^\ast}^-, w^\ast}$
From the Banach-Alaoglu Theorem, $K$ is compact.
From Weak-* Topology is Hausdorff, $K$ is Hausdorff.
Since $K$ is metrizable, $\map \CC K$ is separable from Space of Complex-Valued Continuous Functions on Compact Hausdorff Space is Separable iff Space is Metrizable.
Define $... | Let $X$ be a [[Definition:Normed Vector Space|normed vector space]] such that:
:$\struct {B_{X^\ast}^-, w^\ast}$ is [[Definition:Metrizable Space|metrizable]]
where $X^\ast$ is the [[Definition:Normed Dual Space|normed dual]] of $X$ and $B_{X^\ast}^-$ is the [[Definition:Closed Unit Ball|closed unit ball]] of $X^\as... | Let:
:$K = \struct {B_{X^\ast}^-, w^\ast}$
From the [[Banach-Alaoglu Theorem]], $K$ is [[Definition:Compact Topological Space|compact]].
From [[Weak-* Topology is Hausdorff]], $K$ is [[Definition:Hausdorff Space|Hausdorff]].
Since $K$ is [[Definition:Metrizable Space|metrizable]], $\map \CC K$ is [[Definition:Sepa... | Weak-* Metrizability of Closed Unit Ball in Normed Dual of Normed Vector Space implies Original Space is Separable | https://proofwiki.org/wiki/Weak-*_Metrizability_of_Closed_Unit_Ball_in_Normed_Dual_of_Normed_Vector_Space_implies_Original_Space_is_Separable | https://proofwiki.org/wiki/Weak-*_Metrizability_of_Closed_Unit_Ball_in_Normed_Dual_of_Normed_Vector_Space_implies_Original_Space_is_Separable | [
"Weak-* Topologies",
"Separable Spaces",
"Metrizable Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Metrizable Space",
"Definition:Normed Dual Space",
"Definition:Closed Unit Ball",
"Definition:Separable Space"
] | [
"Banach-Alaoglu Theorem",
"Definition:Compact Topological Space",
"Weak-* Topology is Hausdorff",
"Definition:T2 Space",
"Definition:Metrizable Space",
"Definition:Separable Space",
"Space of Complex-Valued Continuous Functions on Compact Hausdorff Space is Separable iff Space is Metrizable",
"Definit... |
proofwiki-20909 | Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Homeomorphism onto Image | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a normed vector space over $\Bbb F$.
Let $X^\ast$ be the normed dual of $X$.
Let $X^{\ast \ast}$ be the second normed dual of $X$.
Let $w$ be the weak topology on $X$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^{\ast \ast}$.
Let $\iota : X \to X^{\ast \ast}$ be the eval... | From Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Continuous Embedding into Second Normed Dual, $\iota$ is continuous.
From Evaluation Linear Transformation on Normed Vector Space is Linear Isometry, $\iota$ is bijective.
It remains to show that $\iota^{-1} : \struct {\iota X, w^\ast} \to \... | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual]] of $X$.
Let $X^{\ast \ast}$ be the [[Definition:Second Normed Dual|second normed dual]] of $X$.
Let $w$ be the [[Definition:Weak Topo... | From [[Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Continuous Embedding into Second Normed Dual]], $\iota$ is [[Definition:Continuous Mapping|continuous]].
From [[Evaluation Linear Transformation on Normed Vector Space is Linear Isometry]], $\iota$ is [[Definition:Bijection|bijective]].
... | Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Homeomorphism onto Image | https://proofwiki.org/wiki/Evaluation_Linear_Transformation_on_Normed_Vector_Space_is_Weak_to_Weak-*_Homeomorphism_onto_Image | https://proofwiki.org/wiki/Evaluation_Linear_Transformation_on_Normed_Vector_Space_is_Weak_to_Weak-*_Homeomorphism_onto_Image | [
"Evaluation Linear Transformations",
"Weak Topologies on Topological Vector Spaces",
"Weak-* Topologies"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Second Normed Dual",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Weak-* Topology",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Definition:Homeomorphism"
] | [
"Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Continuous Embedding into Second Normed Dual",
"Definition:Continuous Mapping",
"Evaluation Linear Transformation on Normed Vector Space is Linear Isometry",
"Definition:Bijection",
"Definition:Continuous Mapping",
"Continuity in I... |
proofwiki-20910 | Characterization of Continuity of Linear Functional in 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 $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.
Let $X^{\ast \ast}$ be the second normed dual of $X$.
Then a linear functional $\phi : \struct {X^\ast, w^\ast}... | Note that the weak-$\ast$ topology $w^\ast$ is generated as an initial topology by $\set {x^\wedge : x \in X}$.
This result is then given by Continuity of Linear Functionals in Initial Topology on Vector Space Generated by Linear Functionals, taking $F = \set {x^\wedge : x \in X}$.
{{qed}} | 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 $w^\ast$ be the [[Definition:Weak-* Topology|weak-$\ast$ topology]] on $X^\ast$.
Let $X^{... | Note that the [[Definition:Weak-* Topology|weak-$\ast$ topology]] $w^\ast$ is [[Definition:Initial Topology|generated as an initial topology]] by $\set {x^\wedge : x \in X}$.
This result is then given by [[Continuity of Linear Functionals in Initial Topology on Vector Space Generated by Linear Functionals]], taking $... | 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"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Definition:Second Normed Dual",
"Definition:Linear Functional",
"Definition:Continuous Mapping",
"Definition:Evaluation Linear Transformation"
] | [
"Definition:Weak-* Topology",
"Definition:Initial Topology",
"Continuity of Linear Functionals in Weak Topology Induced by Pair of Vector Spaces with Bilinear Mapping/Corollary"
] |
proofwiki-20911 | Normed Dual of Normed Vector Space is Separable iff Closed Unit Ball is Metrizable | Let $\GF \in \set {\R, \C}$.
Let $X$ be a normed vector space over $\GF$.
Let $w$ be the weak topology on $X$.
Let $X^\ast$ be the normed dual space of $X$.
Let $B_X^-$ be the closed unit ball of $X$.
Then $X^\ast$ is separable {{iff}} $\struct {B_X^-, w}$ is metrizable. | Let $X^{\ast \ast}$ be the second normed dual.
Let $\iota : X \to X^{\ast \ast}$ be the evaluation linear transformation. | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $X$.
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Let $B_X^-$ be the [[Definition:C... | Let $X^{\ast \ast}$ be the [[Definition:Second Normed Dual|second normed dual]].
Let $\iota : X \to X^{\ast \ast}$ be the [[Definition:Evaluation Linear Transformation|evaluation linear transformation]]. | Normed Dual of Normed Vector Space is Separable iff Closed Unit Ball is Metrizable | https://proofwiki.org/wiki/Normed_Dual_of_Normed_Vector_Space_is_Separable_iff_Closed_Unit_Ball_is_Metrizable | https://proofwiki.org/wiki/Normed_Dual_of_Normed_Vector_Space_is_Separable_iff_Closed_Unit_Ball_is_Metrizable | [
"Separable Spaces",
"Normed Dual Spaces",
"Metrizable Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Normed Dual Space",
"Definition:Closed Unit Ball",
"Definition:Separable Space",
"Definition:Metrizable Space"
] | [
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation"
] |
proofwiki-20912 | Image of Subset under Composite Relation | Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be relations.
Let $\RR_2 \circ \RR_1 \subseteq S_1 \times T_2$ be the composition of $\RR_1$ and $\RR_2$.
Let $A \subseteq S_1$.
Then:
:$\RR_2 \sqbrk {\RR_1 \sqbrk A \cap S_2} = \paren{\RR_2 \circ \RR_1} \sqbrk A$ | We have:
{{begin-eqn}}
{{eqn | q = \forall z \in T_2
| l = z \in \RR_2 \sqbrk {\RR_1 \sqbrk A \cap S_2}
| o = \leadstoandfrom
| r = \exists y \in \RR_1 \sqbrk A \cap S_2 : \tuple{y, z} \in R_2
| c = {{Defof|Image of Subset under Relation}}
}}
{{eqn | o = \leadstoandfrom
| r = \exists y \in... | Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be [[Definition:Relation|relations]].
Let $\RR_2 \circ \RR_1 \subseteq S_1 \times T_2$ be the [[Definition:Composition of Relations|composition of $\RR_1$ and $\RR_2$]].
Let $A \subseteq S_1$.
Then:
:$\RR_2 \sqbrk {\RR_1 \sqbrk A \cap S_2} = ... | We have:
{{begin-eqn}}
{{eqn | q = \forall z \in T_2
| l = z \in \RR_2 \sqbrk {\RR_1 \sqbrk A \cap S_2}
| o = \leadstoandfrom
| r = \exists y \in \RR_1 \sqbrk A \cap S_2 : \tuple{y, z} \in R_2
| c = {{Defof|Image of Subset under Relation}}
}}
{{eqn | o = \leadstoandfrom
| r = \exists y \in... | Image of Subset under Composite Relation | https://proofwiki.org/wiki/Image_of_Subset_under_Composite_Relation | https://proofwiki.org/wiki/Image_of_Subset_under_Composite_Relation | [
"Composite Relations"
] | [
"Definition:Relation",
"Definition:Composition of Relations"
] | [] |
proofwiki-20913 | Image of Element under Composite Relation | Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be relations.
Let $\RR_2 \circ \RR_1 \subseteq S_1 \times T_2$ be the composition of $\RR_1$ and $\RR_2$.
Let $x \in S_1$.
Then:
:$\RR_2 \sqbrk {\map {\RR_1} x \cap S_2} = \map {\paren{\RR_2 \circ \RR_1}} x$ | We have:
{{begin-eqn}}
{{eqn | l = \RR_2 \sqbrk {\map {\RR_1} x \cap S_2}
| r = \RR_2 \sqbrk {\RR_1 \sqbrk {\set x} \cap S_2}
| c = Image of Singleton under Relation
}}
{{eqn | r = \paren{\RR_2 \circ \RR_1} \sqbrk {\set x}
| c = Image of Subset under Composite Relation
}}
{{eqn | r = \map {\paren{\RR_... | Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be [[Definition:Relation|relations]].
Let $\RR_2 \circ \RR_1 \subseteq S_1 \times T_2$ be the [[Definition:Composition of Relations|composition of $\RR_1$ and $\RR_2$]].
Let $x \in S_1$.
Then:
:$\RR_2 \sqbrk {\map {\RR_1} x \cap S_2} = \map {... | We have:
{{begin-eqn}}
{{eqn | l = \RR_2 \sqbrk {\map {\RR_1} x \cap S_2}
| r = \RR_2 \sqbrk {\RR_1 \sqbrk {\set x} \cap S_2}
| c = [[Image of Singleton under Relation]]
}}
{{eqn | r = \paren{\RR_2 \circ \RR_1} \sqbrk {\set x}
| c = [[Image of Subset under Composite Relation]]
}}
{{eqn | r = \map {\pa... | Image of Element under Composite Relation | https://proofwiki.org/wiki/Image_of_Element_under_Composite_Relation | https://proofwiki.org/wiki/Image_of_Element_under_Composite_Relation | [
"Composite Relations"
] | [
"Definition:Relation",
"Definition:Composition of Relations"
] | [
"Image of Singleton under Relation",
"Image of Subset under Composite Relation",
"Image of Singleton under Relation"
] |
proofwiki-20914 | Image of Subset under Composite Relation with Common Codomain and Domain | Let $\RR_1 \subseteq S \times T$ and $\RR_2 \subseteq T \times U$ be relations.
Let $\RR_2 \circ \RR_1 \subseteq S \times U$ be the composition of $\RR_1$ and $\RR_2$.
Let $A \subseteq S$.
Then:
:$\RR_2 \sqbrk {\RR_1 \sqbrk A} = \paren{\RR_2 \circ \RR_1} \sqbrk A$ | We have:
{{begin-eqn}}
{{eqn | l = \RR_1 \sqbrk A
| o = \subseteq
| r = T
| c = Image is Subset of Codomain
}}
{{eqn | ll = \leadsto
| l = \RR_1 \sqbrk A
| r = \RR_1 \sqbrk A \cap T
| c = Intersection with Subset is Subset
}}
{{eqn | ll = \leadsto
| l = \RR_2 \sqbrk {\RR_1 \sqb... | Let $\RR_1 \subseteq S \times T$ and $\RR_2 \subseteq T \times U$ be [[Definition:Relation|relations]].
Let $\RR_2 \circ \RR_1 \subseteq S \times U$ be the [[Definition:Composition of Relations|composition of $\RR_1$ and $\RR_2$]].
Let $A \subseteq S$.
Then:
:$\RR_2 \sqbrk {\RR_1 \sqbrk A} = \paren{\RR_2 \circ \RR_... | We have:
{{begin-eqn}}
{{eqn | l = \RR_1 \sqbrk A
| o = \subseteq
| r = T
| c = [[Image is Subset of Codomain]]
}}
{{eqn | ll = \leadsto
| l = \RR_1 \sqbrk A
| r = \RR_1 \sqbrk A \cap T
| c = [[Intersection with Subset is Subset]]
}}
{{eqn | ll = \leadsto
| l = \RR_2 \sqbrk {\R... | Image of Subset under Composite Relation with Common Codomain and Domain | https://proofwiki.org/wiki/Image_of_Subset_under_Composite_Relation_with_Common_Codomain_and_Domain | https://proofwiki.org/wiki/Image_of_Subset_under_Composite_Relation_with_Common_Codomain_and_Domain | [
"Composite Relations"
] | [
"Definition:Relation",
"Definition:Composition of Relations"
] | [
"Image is Subset of Codomain",
"Intersection with Subset is Subset",
"Image of Subset under Composite Relation"
] |
proofwiki-20915 | Image of Element under Composite Relation with Common Codomain and Domain | Let $\RR_1 \subseteq S \times T$ and $\RR_2 \subseteq T \times U$ be relations.
Let $\RR_2 \circ \RR_1 \subseteq S \times U$ be the composition of $\RR_1$ and $\RR_2$.
Let $x \in S$.
Then:
:$\RR_2 \sqbrk {\map {\RR_1} x} = \map {\paren{\RR_2 \circ \RR_1}} x$ | We have:
{{begin-eqn}}
{{eqn | l = \RR_2 \sqbrk {\map {\RR_1} x}
| r = \RR_2 \sqbrk {\RR_1 \sqbrk {\set x} }
| c = Image of Singleton under Relation
}}
{{eqn | r = \paren{\RR_2 \circ \RR_1} \sqbrk {\set x}
| c = Image of Subset under Composite Relation with Common Codomain and Domain
}}
{{eqn | r = \m... | Let $\RR_1 \subseteq S \times T$ and $\RR_2 \subseteq T \times U$ be [[Definition:Relation|relations]].
Let $\RR_2 \circ \RR_1 \subseteq S \times U$ be the [[Definition:Composition of Relations|composition of $\RR_1$ and $\RR_2$]].
Let $x \in S$.
Then:
:$\RR_2 \sqbrk {\map {\RR_1} x} = \map {\paren{\RR_2 \circ \RR_... | We have:
{{begin-eqn}}
{{eqn | l = \RR_2 \sqbrk {\map {\RR_1} x}
| r = \RR_2 \sqbrk {\RR_1 \sqbrk {\set x} }
| c = [[Image of Singleton under Relation]]
}}
{{eqn | r = \paren{\RR_2 \circ \RR_1} \sqbrk {\set x}
| c = [[Image of Subset under Composite Relation with Common Codomain and Domain]]
}}
{{eqn ... | Image of Element under Composite Relation with Common Codomain and Domain | https://proofwiki.org/wiki/Image_of_Element_under_Composite_Relation_with_Common_Codomain_and_Domain | https://proofwiki.org/wiki/Image_of_Element_under_Composite_Relation_with_Common_Codomain_and_Domain | [
"Composite Relations"
] | [
"Definition:Relation",
"Definition:Composition of Relations"
] | [
"Image of Singleton under Relation",
"Image of Subset under Composite Relation with Common Codomain and Domain",
"Image of Singleton under Relation"
] |
proofwiki-20916 | Normed Vector Space is Reflexive iff Weak and Weak-* Topologies on Normed Dual coincide | Let $X$ be a normed vector space.
Let $X^\ast$ be the normed dual of $X$.
Let $w$ be the weak topology on $X^\ast$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.
Then $X$ is reflexive {{iff}} $\struct {X^\ast, w} = \struct {X^\ast, w^\ast}$. | === Necessary Condition ===
Suppose that $X$ is reflexive.
Then for each $\Phi \in X^{\ast \ast}$ there exists $x \in X$ such that $\Phi = x^\wedge$.
Conversely, by Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual, we have $x^\wedge \in X^{\ast \ast}$ for... | Let $X$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual]] of $X$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $X^\ast$.
Let $w^\ast$ be the [[Definition:Weak-* Topology|weak-$\ast$ topology]] on... | === Necessary Condition ===
Suppose that $X$ is [[Definition:Reflexive Space|reflexive]].
Then for each $\Phi \in X^{\ast \ast}$ there exists $x \in X$ such that $\Phi = x^\wedge$.
Conversely, by [[Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual]], we... | Normed Vector Space is Reflexive iff Weak and Weak-* Topologies on Normed Dual coincide | https://proofwiki.org/wiki/Normed_Vector_Space_is_Reflexive_iff_Weak_and_Weak-*_Topologies_on_Normed_Dual_coincide | https://proofwiki.org/wiki/Normed_Vector_Space_is_Reflexive_iff_Weak_and_Weak-*_Topologies_on_Normed_Dual_coincide | [
"Weak Topologies on Topological Vector Spaces",
"Weak-* Topologies",
"Reflexive Spaces",
"Normed Dual Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Weak-* Topology",
"Definition:Reflexive Space"
] | [
"Definition:Reflexive Space",
"Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual",
"Definition:Initial Topology",
"Definition:Initial Topology",
"Definition:Reflexive Space"
] |
proofwiki-20917 | Normed Vector Space is Reflexive iff Closed Unit Ball in Original Space is Mapped to Closed Unit Ball in Second Dual | Let $X$ be a normed vector space.
Let $X^\ast$ be the normed dual of $X$.
Let $X^{\ast \ast}$ be the second norm dual.
Let $\iota : X \to X^{\ast \ast}$ be the evaluation linear transformation.
Let $B_X^-$ be the closed unit ball of $X$.
Let $B_{X^{\ast \ast} }^-$ be the closed unit ball of $X^{\ast \ast}$.
Then $X$ ... | === Necessary Condition ===
Suppose that $X$ is reflexive.
Then $\iota : X \to X^{\ast \ast}$ is an isometric isomorphism.
So from Injective Linear Transformation between Normed Vector Spaces sends Closed Unit Ball to Closed Unit Ball iff Isometric Isomorphism, we have that $\iota B_X^- = B_{X^{\ast \ast} }^-$.
{{qed|... | Let $X$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual]] of $X$.
Let $X^{\ast \ast}$ be the [[Definition:Second Normed Dual|second norm dual]].
Let $\iota : X \to X^{\ast \ast}$ be the [[Definition:Evaluation Linear Transformation|evaluati... | === Necessary Condition ===
Suppose that $X$ is [[Definition:Reflexive Space|reflexive]].
Then $\iota : X \to X^{\ast \ast}$ is an [[Definition:Isometric Isomorphism on Normed Vector Space|isometric isomorphism]].
So from [[Injective Linear Transformation between Normed Vector Spaces sends Closed Unit Ball to Closed... | Normed Vector Space is Reflexive iff Closed Unit Ball in Original Space is Mapped to Closed Unit Ball in Second Dual | https://proofwiki.org/wiki/Normed_Vector_Space_is_Reflexive_iff_Closed_Unit_Ball_in_Original_Space_is_Mapped_to_Closed_Unit_Ball_in_Second_Dual | https://proofwiki.org/wiki/Normed_Vector_Space_is_Reflexive_iff_Closed_Unit_Ball_in_Original_Space_is_Mapped_to_Closed_Unit_Ball_in_Second_Dual | [
"Reflexive Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation",
"Definition:Closed Unit Ball",
"Definition:Closed Unit Ball",
"Definition:Reflexive Space"
] | [
"Definition:Reflexive Space",
"Definition:Isometric Isomorphism/Normed Vector Space",
"Injective Linear Transformation between Normed Vector Spaces sends Closed Unit Ball to Closed Unit Ball iff Isometric Isomorphism",
"Definition:Isometric Isomorphism/Normed Vector Space",
"Injective Linear Transformation ... |
proofwiki-20918 | Kakutani's Theorem | Let $X$ be a normed vector space.
Let $w$ be the weak topology on $X$.
Let $B_X^-$ be the closed unit ball in $X$.
Let $X^{\ast \ast}$ be the second normed dual of $X$.
Then $X$ is reflexive {{iff}} $\struct {B_X^-, w}$ is compact. | Let $B_{X^{\ast \ast} }^-$ be the closed unit ball in $X^{\ast \ast}$.
Let $w^\ast$ be the weak-* topology on $X^{\ast \ast}$.
Let $\iota : X \to X^{\ast \ast}$ be the evaluation linear transformation. | Let $X$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $X$.
Let $B_X^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $X$.
Let $X^{\ast \ast}$ be the [[Definition:Second Normed Dual|second normed dua... | Let $B_{X^{\ast \ast} }^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $X^{\ast \ast}$.
Let $w^\ast$ be the [[Definition:Weak-* Topology|weak-* topology]] on $X^{\ast \ast}$.
Let $\iota : X \to X^{\ast \ast}$ be the [[Definition:Evaluation Linear Transformation|evaluation linear transformation]]. | Kakutani's Theorem | https://proofwiki.org/wiki/Kakutani's_Theorem | https://proofwiki.org/wiki/Kakutani's_Theorem | [
"Normed Vector Spaces",
"Reflexive Spaces",
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Closed Unit Ball",
"Definition:Second Normed Dual",
"Definition:Reflexive Space",
"Definition:Compact Topological Space"
] | [
"Definition:Closed Unit Ball",
"Definition:Weak-* Topology",
"Definition:Evaluation Linear Transformation"
] |
proofwiki-20919 | Closed Set in Coarser Topology is Closed in Finer Topology | Let $X$ be a set.
Let $\tau_1$ and $\tau_2$ be topologies on $X$ such that $\tau_1 \subseteq \tau_2$.
That is, $\tau_1$ is coarser than $\tau_2$.
Let $C$ be a closed set in $\struct {X, \tau_1}$.
Then $C$ is closed in $\struct {X, \tau_2}$. | Let $C$ be a closed set in $\struct {X, \tau_1}$.
Then by definition of closed set:
:$X \setminus C$ is open in $\struct {X, \tau_1}$.
Hence by definition of open set:
:$X \setminus C \in \tau_1$
By definition of subset:
:$X \setminus C \in \tau_2$
So by definition of open set:
:$X \setminus C$ is open in $\struct {X, ... | Let $X$ be a [[Definition:Set|set]].
Let $\tau_1$ and $\tau_2$ be [[Definition:Topology|topologies]] on $X$ such that $\tau_1 \subseteq \tau_2$.
That is, $\tau_1$ is [[Definition:Coarser Topology|coarser]] than $\tau_2$.
Let $C$ be a [[Definition:Closed Set (Topology)|closed set]] in $\struct {X, \tau_1}$.
Then... | Let $C$ be a [[Definition:Closed Set (Topology)|closed set]] in $\struct {X, \tau_1}$.
Then by definition of [[Definition:Closed Set (Topology)|closed set]]:
:$X \setminus C$ is [[Definition:Open Set (Topology)|open]] in $\struct {X, \tau_1}$.
Hence by definition of [[Definition:Open Set (Topology)|open set]]:
:$X \s... | Closed Set in Coarser Topology is Closed in Finer Topology | https://proofwiki.org/wiki/Closed_Set_in_Coarser_Topology_is_Closed_in_Finer_Topology | https://proofwiki.org/wiki/Closed_Set_in_Coarser_Topology_is_Closed_in_Finer_Topology | [
"Closed Sets"
] | [
"Definition:Set",
"Definition:Topology",
"Definition:Coarser Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology"
] | [
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Category:Closed Sets"
] |
proofwiki-20920 | Multiple of Vector in Topological Vector Space Converges | Let $K$ be a topological field.
Let $\struct {X, \tau}$ be a topological vector space.
Let $x \in X$.
Let $\lambda \in K$.
Let $\sequence {\lambda_n}_{n \mathop \in \N}$ be a sequence in $K$ such that $\lambda_n \to \lambda$.
Then:
:$\lambda_n x \to \lambda x$ | From the definition of a topological vector space, the map $f : K \times \struct {X, \tau} \to X$ defined by:
:$\map f {\lambda, y} = \lambda y$
is continuous.
From Horizontal Section of Continuous Function is Continuous, we therefore have the map $c_x : X \to X$ defined by:
:$\map {c_x} \lambda = \lambda x$
is con... | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]].
Let $x \in X$.
Let $\lambda \in K$.
Let $\sequence {\lambda_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $K$ such that $\lambda_n \to ... | From the definition of a [[Definition:Topological Vector Space|topological vector space]], the map $f : K \times \struct {X, \tau} \to X$ defined by:
:$\map f {\lambda, y} = \lambda y$
is [[Definition:Continuous Mapping|continuous]].
From [[Horizontal Section of Continuous Function is Continuous]], we therefore ha... | Multiple of Vector in Topological Vector Space Converges | https://proofwiki.org/wiki/Multiple_of_Vector_in_Topological_Vector_Space_Converges | https://proofwiki.org/wiki/Multiple_of_Vector_in_Topological_Vector_Space_Converges | [
"Topological Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Sequence"
] | [
"Definition:Topological Vector Space",
"Definition:Continuous Mapping",
"Horizontal Section of Continuous Function is Continuous",
"Definition:Continuous Mapping",
"Continuous Mapping is Sequentially Continuous",
"Definition:Sequential Continuity",
"Category:Topological Vector Spaces"
] |
proofwiki-20921 | Absorbing Set in Vector Space contains Zero Vector | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $A \subseteq X$ be an absorbing subset of $X$.
Then:
:$\mathbf 0_X \in A$
where $\mathbf 0_X$ denotes the zero vector in $X$. | From the definition of an absorbing subset, there exists $t \in \R_{>0}$ such that ${\mathbf 0}_X \in t A$.
So $\mathbf 0_X \in A$.
{{qed}}
Category:Absorbing Sets
t6y5k1oruhc7kxw5uz1l15faj95w7qp | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $A \subseteq X$ be an [[Definition:Absorbing Set|absorbing subset]] of $X$.
Then:
:$\mathbf 0_X \in A$
where $\mathbf 0_X$ denotes the [[Definition:Zero Vector|zero vector]] in $X$. | From the definition of an [[Definition:Absorbing Set|absorbing subset]], there exists $t \in \R_{>0}$ such that ${\mathbf 0}_X \in t A$.
So $\mathbf 0_X \in A$.
{{qed}}
[[Category:Absorbing Sets]]
t6y5k1oruhc7kxw5uz1l15faj95w7qp | Absorbing Set in Vector Space contains Zero Vector | https://proofwiki.org/wiki/Absorbing_Set_in_Vector_Space_contains_Zero_Vector | https://proofwiki.org/wiki/Absorbing_Set_in_Vector_Space_contains_Zero_Vector | [
"Absorbing Sets"
] | [
"Definition:Vector Space",
"Definition:Absorbing Set",
"Definition:Zero Vector"
] | [
"Definition:Absorbing Set",
"Category:Absorbing Sets"
] |
proofwiki-20922 | Image under Left-Total Relation is Empty iff Subset is Empty | Let $\RR \subseteq S \times T$ be a left-total relation.
Let $A \subseteq S$.
Then:
:$\RR \sqbrk A = \O$ {{iff}} $A = \O$ | === Necessary Condition ===
We prove the contrapositive statement:
:$A \ne \O \implies \RR \sqbrk A \ne \O$
Let $s \in A$.
By definition of left-total relation:
:$\exists t \in T : \tuple{s, t} \in \RR$
By definition of image:
:$\exists t \in T : t \in \RR \sqbrk A$
Hence:
:$\RR \sqbrk A \ne \O$
The result follows from... | Let $\RR \subseteq S \times T$ be a [[Definition:Left-Total Relation|left-total relation]].
Let $A \subseteq S$.
Then:
:$\RR \sqbrk A = \O$ {{iff}} $A = \O$ | === Necessary Condition ===
We prove the [[Definition:Contrapositive Statement|contrapositive statement]]:
:$A \ne \O \implies \RR \sqbrk A \ne \O$
Let $s \in A$.
By definition of [[Definition:Left-Total Relation|left-total relation]]:
:$\exists t \in T : \tuple{s, t} \in \RR$
By definition of [[Definition:Image ... | Image under Left-Total Relation is Empty iff Subset is Empty | https://proofwiki.org/wiki/Image_under_Left-Total_Relation_is_Empty_iff_Subset_is_Empty | https://proofwiki.org/wiki/Image_under_Left-Total_Relation_is_Empty_iff_Subset_is_Empty | [
"Composite Relations"
] | [
"Definition:Left-Total Relation"
] | [
"Definition:Contrapositive Statement",
"Definition:Left-Total Relation",
"Definition:Image (Set Theory)/Relation/Subset",
"Rule of Transposition"
] |
proofwiki-20923 | Minkowski Functional of Convex Absorbing Set is Positive Homogeneous | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $A$ be a convex absorbing set.
Let $\mu_A$ be the Minkowski functional of $A$.
Then $\mu_A$ is positive homogeneous.
That is, for each $t \ge 0$ we have:
:$\map {\mu_A} {t x} = t \map {\mu_A} x$ | From Absorbing Set in Vector Space contains Zero Vector, we have ${\mathbf 0}_X \in A$.
So $\map {\mu_A} {\mathbf 0_X} = 0$.
So the claim clearly holds for $t = 0$.
Now take $t > 0$.
We argue that:
:$\set {s > 0 : t x \in s C} = t \set {s > 0 : x \in s C}$
We have:
:$s \in \set {s > 0 : t x \in s C}$
{{iff}} $s > 0... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $A$ be a [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Absorbing Set|absorbing set]].
Let $\mu_A$ be the [[Definition:Minkowski Functional|Minkowski functional]] of $A$.
Then $\mu_A$ is [[Definition... | From [[Absorbing Set in Vector Space contains Zero Vector]], we have ${\mathbf 0}_X \in A$.
So $\map {\mu_A} {\mathbf 0_X} = 0$.
So the claim clearly holds for $t = 0$.
Now take $t > 0$.
We argue that:
:$\set {s > 0 : t x \in s C} = t \set {s > 0 : x \in s C}$
We have:
:$s \in \set {s > 0 : t x \in s C}$
{{... | Minkowski Functional of Convex Absorbing Set is Positive Homogeneous | https://proofwiki.org/wiki/Minkowski_Functional_of_Convex_Absorbing_Set_is_Positive_Homogeneous | https://proofwiki.org/wiki/Minkowski_Functional_of_Convex_Absorbing_Set_is_Positive_Homogeneous | [
"Minkowski Functionals"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Absorbing Set",
"Definition:Minkowski Functional",
"Definition:Positive Homogeneous"
] | [
"Absorbing Set in Vector Space contains Zero Vector",
"Multiple of Infimum",
"Category:Minkowski Functionals"
] |
proofwiki-20924 | Convex Subset of Topological Vector Space containing Zero Vector in Interior is Absorbing Set | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $C \subseteq X$ be a convex set such that ${\mathbf 0}_X \in C^\circ$, where $C^\circ$ denotes the interior of $C$.
Then $C$ is an absorbing set. | Let $x \in X$.
Let $V$ be an open neighborhood of ${\mathbf 0}_X$ contained in $C$.
Then from Multiple of Vector in Topological Vector Space Converges, we have:
:$\ds \frac x n \to {\mathbf 0}_X$
So by the definition of a convergent sequence, for some $N \in \N$ we have:
:$\ds \frac x N \in V$
Then:
:$\ds x \in N ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $C \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex set]] such that ${\mathbf 0}_X \in C^\circ$, where $C^\circ$ denotes the [[Definition:Interior|interior]] of $... | Let $x \in X$.
Let $V$ be an [[Definition:Open Neighborhood|open neighborhood]] of ${\mathbf 0}_X$ contained in $C$.
Then from [[Multiple of Vector in Topological Vector Space Converges]], we have:
:$\ds \frac x n \to {\mathbf 0}_X$
So by the definition of a [[Definition:Convergent Sequence|convergent sequence]]... | Convex Subset of Topological Vector Space containing Zero Vector in Interior is Absorbing Set | https://proofwiki.org/wiki/Convex_Subset_of_Topological_Vector_Space_containing_Zero_Vector_in_Interior_is_Absorbing_Set | https://proofwiki.org/wiki/Convex_Subset_of_Topological_Vector_Space_containing_Zero_Vector_in_Interior_is_Absorbing_Set | [
"Convex Sets (Vector Spaces)",
"Absorbing Sets",
"Topological Vector Spaces",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Topological Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Interior",
"Definition:Absorbing Set"
] | [
"Definition:Open Neighborhood",
"Multiple of Vector in Topological Vector Space Converges",
"Definition:Convergent Sequence",
"Characterization of Convex Absorbing Set in Vector Space",
"Definition:Absorbing Set",
"Category:Absorbing Sets",
"Category:Topological Vector Spaces",
"Category:Convex Sets (... |
proofwiki-20925 | Minkowski Functional of Open Convex Set containing Zero Vector in Topological Vector Space recovers Set | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $C \subseteq X$ be an open convex set with ${\mathbf 0}_X \in C$.
From Convex Subset of Topological Vector Space containing Zero Vector in Interior is Absorbing Set, $C$ is absorbing.
Let $\mu_C$ be the Minkowski functi... | From Convex Absorbing Set contained between Sets in terms of Minkowski Functional, we have:
:$\set {x \in X : \map {\mu_C} x < 1} \subseteq C$
Conversely, suppose that $x \in C$.
From Multiple of Vector in Topological Vector Space Converges, we have:
:$\paren {1 + \dfrac 1 n} x \to x$
From the definition of a conver... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $C \subseteq X$ be an [[Definition:Open Set|open]] [[Definition:Convex Set (Vector Space)|convex set]] with ${\mathbf 0}_X \in C$.
From [[Convex Subset of Topological Vector Sp... | From [[Convex Absorbing Set contained between Sets in terms of Minkowski Functional]], we have:
:$\set {x \in X : \map {\mu_C} x < 1} \subseteq C$
Conversely, suppose that $x \in C$.
From [[Multiple of Vector in Topological Vector Space Converges]], we have:
:$\paren {1 + \dfrac 1 n} x \to x$
From the definiti... | Minkowski Functional of Open Convex Set containing Zero Vector in Topological Vector Space recovers Set | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Convex_Set_containing_Zero_Vector_in_Topological_Vector_Space_recovers_Set | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Convex_Set_containing_Zero_Vector_in_Topological_Vector_Space_recovers_Set | [
"Minkowski Functionals"
] | [
"Definition:Topological Vector Space",
"Definition:Open Set",
"Definition:Convex Set (Vector Space)",
"Convex Subset of Topological Vector Space containing Zero Vector in Interior is Absorbing Set",
"Definition:Absorbing Set",
"Definition:Minkowski Functional"
] | [
"Convex Absorbing Set contained between Sets in terms of Minkowski Functional",
"Multiple of Vector in Topological Vector Space Converges",
"Definition:Convergent Sequence",
"Definition:Minkowski Functional",
"Category:Minkowski Functionals"
] |
proofwiki-20926 | Convex Absorbing Set contained between Sets in terms of Minkowski Functional | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $C$ be a convex absorbing set.
Let $\mu_C$ be the Minkowski functional of $C$.
Then we have:
:$\set {x \in X : \map {\mu_C} x < 1} \subseteq C \subseteq \set {x \in X : \map {\mu_C} x \le 1}$ | Let $x \in X$ be such that $\map {\mu_C} x < 1$.
Then:
:$\inf \set {t > 0 : \dfrac x t \in C} < 1$
So there exists $t < 1$ such that $x \in t C$.
From Absorbing Set in Vector Space contains Zero Vector, we have that ${\mathbf 0}_X \in C$.
So, we have, since $C$ is convex:
:$x = x + \paren {1 - t} {\mathbf 0}_X \in t... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $C$ be a [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Absorbing Set|absorbing set]].
Let $\mu_C$ be the [[Definition:Minkowski Functional|Minkowski functional]]... | Let $x \in X$ be such that $\map {\mu_C} x < 1$.
Then:
:$\inf \set {t > 0 : \dfrac x t \in C} < 1$
So there exists $t < 1$ such that $x \in t C$.
From [[Absorbing Set in Vector Space contains Zero Vector]], we have that ${\mathbf 0}_X \in C$.
So, we have, since $C$ is [[Definition:Convex Set (Vector Space)|conve... | Convex Absorbing Set contained between Sets in terms of Minkowski Functional | https://proofwiki.org/wiki/Convex_Absorbing_Set_contained_between_Sets_in_terms_of_Minkowski_Functional | https://proofwiki.org/wiki/Convex_Absorbing_Set_contained_between_Sets_in_terms_of_Minkowski_Functional | [
"Minkowski Functionals"
] | [
"Definition:Topological Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Absorbing Set",
"Definition:Minkowski Functional"
] | [
"Absorbing Set in Vector Space contains Zero Vector",
"Definition:Convex Set (Vector Space)",
"Category:Minkowski Functionals"
] |
proofwiki-20927 | Real Linear Functional is Real Part of Unique Linear Functional | Let $X$ be a vector space over $\C$.
Let $g : X \to \R$ be a $\R$-linear functional.
Then there exists a unique $\C$-linear functional $f : X \to \C$ such that:
:$\map g x = \map \Re {\map f x}$
for each $x \in X$.
Further:
:$\map f x = \map g x - i \map g {i x}$ | Define $f : X \to \C$ by:
:$\map f x = \map g x - i \map g {i x}$
for each $x \in X$.
Then for $\lambda, \mu \in \R$ and $x, y \in X$ we have:
{{begin-eqn}}
{{eqn | l = \map f {\lambda x + \mu y}
| r = \map g {\lambda x + \mu y} - i \map g {i \paren {\lambda x + \mu y} }
}}
{{eqn | r = \lambda \map g x + \mu \map... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\C$.
Let $g : X \to \R$ be a [[Definition:Linear Functional|$\R$-linear functional]].
Then there exists a unique [[Definition:Linear Functional|$\C$-linear functional]] $f : X \to \C$ such that:
:$\map g x = \map \Re {\map f x}$
for each $x \in X$.
F... | Define $f : X \to \C$ by:
:$\map f x = \map g x - i \map g {i x}$
for each $x \in X$.
Then for $\lambda, \mu \in \R$ and $x, y \in X$ we have:
{{begin-eqn}}
{{eqn | l = \map f {\lambda x + \mu y}
| r = \map g {\lambda x + \mu y} - i \map g {i \paren {\lambda x + \mu y} }
}}
{{eqn | r = \lambda \map g x + \mu \... | Real Linear Functional is Real Part of Unique Linear Functional | https://proofwiki.org/wiki/Real_Linear_Functional_is_Real_Part_of_Unique_Linear_Functional | https://proofwiki.org/wiki/Real_Linear_Functional_is_Real_Part_of_Unique_Linear_Functional | [
"Linear Functionals"
] | [
"Definition:Vector Space",
"Definition:Linear Functional",
"Definition:Linear Functional"
] | [
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Linear Functional on Complex Vector Space is Uniquely Determined by Real Part",
"Category:Linear Functionals"
] |
proofwiki-20928 | Realification of Normed Dual is Isometrically Isomorphic to the Normed Dual of Realification | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\C$.
Let $\struct {X_\R, \norm {\, \cdot \,} }$ be the realification of $\struct {X, \norm {\, \cdot \,} }$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual of $\struct {X, \norm {\, \cdot \,} }$.
Let $\struct {X^\ast_\R, \... | We first need to show that if $f \in X^\ast$, then $T f = \map \Re f \in X^\ast_\R$.
From Real Part of Linear Functional is Linear Functional, we have that $\map \Re f : X \to \R$ is $\R$-linear.
We just need to show that $\map \Re f$ is bounded.
We have for each $x \in X$:
{{begin-eqn}}
{{eqn | l = \cmod {\map \Re {\... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\C$.
Let $\struct {X_\R, \norm {\, \cdot \,} }$ be the [[Definition:Realification of Complex Vector Space|realification]] of $\struct {X, \norm {\, \cdot \,} }$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast... | We first need to show that if $f \in X^\ast$, then $T f = \map \Re f \in X^\ast_\R$.
From [[Real Part of Linear Functional is Linear Functional]], we have that $\map \Re f : X \to \R$ is [[Definition:Linear Functional|$\R$-linear]].
We just need to show that $\map \Re f$ is [[Definition:Bounded Linear Functional|boun... | Realification of Normed Dual is Isometrically Isomorphic to the Normed Dual of Realification | https://proofwiki.org/wiki/Realification_of_Normed_Dual_is_Isometrically_Isomorphic_to_the_Normed_Dual_of_Realification | https://proofwiki.org/wiki/Realification_of_Normed_Dual_is_Isometrically_Isomorphic_to_the_Normed_Dual_of_Realification | [
"Realifications of Complex Vector Spaces",
"Normed Dual Spaces",
"Isometric Isomorphisms (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Realification of Complex Vector Space",
"Definition:Normed Dual Space",
"Definition:Normed Dual Space",
"Definition:Realification of Complex Vector Space",
"Definition:Isometric Isomorphism/Normed Vector Space",
"Definition:Isometric Isomorphism/Normed Vecto... | [
"Real Part of Linear Functional is Linear Functional",
"Definition:Linear Functional",
"Definition:Bounded Linear Functional",
"Fundamental Property of Norm on Bounded Linear Functional",
"Definition:Bounded Linear Functional",
"Definition:Sequence",
"Definition:Linear Functional",
"Definition:Surject... |
proofwiki-20929 | Primitive of Reciprocal of a x squared + b by Root of c x squared + d | :$\ds \int \dfrac {\d x} {\paren {a x^2 + b} \sqrt {c x^2 + d} } = \begin {cases}
\dfrac 1 {\sqrt {b \paren {a d - b c} } } \arctan \dfrac {x \sqrt {a d - b c} } {\sqrt {b \paren {c x^2 + d} } } + C & : a d > b c \\
\dfrac 1 {2 \sqrt {b \paren {a d - b c} } } \ln \size {\dfrac {\sqrt {b \paren {c x^2 + d} } + x \sqrt {... | === Case $1$: $a d > b c$ ===
{{:Primitive of Reciprocal of a x squared + b by Root of c x squared + d/a d greater than b c}}{{qed|lemma}} | :$\ds \int \dfrac {\d x} {\paren {a x^2 + b} \sqrt {c x^2 + d} } = \begin {cases}
\dfrac 1 {\sqrt {b \paren {a d - b c} } } \arctan \dfrac {x \sqrt {a d - b c} } {\sqrt {b \paren {c x^2 + d} } } + C & : a d > b c \\
\dfrac 1 {2 \sqrt {b \paren {a d - b c} } } \ln \size {\dfrac {\sqrt {b \paren {c x^2 + d} } + x \sqrt {... | === [[Primitive of Reciprocal of a x squared + b by Root of c x squared + d/a d greater than b c|Case $1$: $a d > b c$]] ===
{{:Primitive of Reciprocal of a x squared + b by Root of c x squared + d/a d greater than b c}}{{qed|lemma}} | Primitive of Reciprocal of a x squared + b by Root of c x squared + d | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_+_b_by_Root_of_c_x_squared_+_d | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_+_b_by_Root_of_c_x_squared_+_d | [
"Primitive of Reciprocal of a x squared + b by Root of c x squared + d",
"Primitives of Roots of Quadratic Functions"
] | [] | [
"Primitive of Reciprocal of a x squared + b by Root of c x squared + d/a d greater than b c"
] |
proofwiki-20930 | Primitive of Reciprocal of a x squared + b by Root of c x squared + d/a d greater than b c | :$\ds \int \dfrac {\d x} {\paren {a x^2 + b} \sqrt {c x^2 + d} } = \dfrac 1 {\sqrt {b \paren {a d - b c} } } \arctan \dfrac {x \sqrt {a d - b c} } {\sqrt {b \paren {c x^2 + d} } } + C$ | === Lemma ===
{{:Primitive of Reciprocal of a x squared + b by Root of c x squared + d/Lemma}}
{{ProofWanted}} | :$\ds \int \dfrac {\d x} {\paren {a x^2 + b} \sqrt {c x^2 + d} } = \dfrac 1 {\sqrt {b \paren {a d - b c} } } \arctan \dfrac {x \sqrt {a d - b c} } {\sqrt {b \paren {c x^2 + d} } } + C$ | === [[Primitive of Reciprocal of a x squared + b by Root of c x squared + d/Lemma|Lemma]] ===
{{:Primitive of Reciprocal of a x squared + b by Root of c x squared + d/Lemma}}
{{ProofWanted}} | Primitive of Reciprocal of a x squared + b by Root of c x squared + d/a d greater than b c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_+_b_by_Root_of_c_x_squared_+_d/a_d_greater_than_b_c | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_+_b_by_Root_of_c_x_squared_+_d/a_d_greater_than_b_c | [
"Primitive of Reciprocal of a x squared + b by Root of c x squared + d"
] | [] | [
"Primitive of Reciprocal of a x squared + b by Root of c x squared + d/Lemma"
] |
proofwiki-20931 | Primitive of Reciprocal of a x squared + b by Root of c x squared + d/b c greater than a d | :$\ds \int \dfrac {\d x} {\paren {a x^2 + b} \sqrt {c x^2 + d} } = \dfrac 1 {2 \sqrt {b \paren {a d - b c} } } \ln \size {\dfrac {\sqrt {b \paren {c x^2 + d} } + x \sqrt {b c - a d} } {\sqrt {b \paren {c x^2 + d} } - x \sqrt {b c - a d} } } + C$ | === Lemma ===
{{:Primitive of Reciprocal of a x squared + b by Root of c x squared + d/Lemma}}
{{ProofWanted}} | :$\ds \int \dfrac {\d x} {\paren {a x^2 + b} \sqrt {c x^2 + d} } = \dfrac 1 {2 \sqrt {b \paren {a d - b c} } } \ln \size {\dfrac {\sqrt {b \paren {c x^2 + d} } + x \sqrt {b c - a d} } {\sqrt {b \paren {c x^2 + d} } - x \sqrt {b c - a d} } } + C$ | === [[Primitive of Reciprocal of a x squared + b by Root of c x squared + d/Lemma|Lemma]] ===
{{:Primitive of Reciprocal of a x squared + b by Root of c x squared + d/Lemma}}
{{ProofWanted}} | Primitive of Reciprocal of a x squared + b by Root of c x squared + d/b c greater than a d | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_+_b_by_Root_of_c_x_squared_+_d/b_c_greater_than_a_d | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_+_b_by_Root_of_c_x_squared_+_d/b_c_greater_than_a_d | [
"Primitive of Reciprocal of a x squared + b by Root of c x squared + d"
] | [] | [
"Primitive of Reciprocal of a x squared + b by Root of c x squared + d/Lemma"
] |
proofwiki-20932 | Minkowski Functional of Convex Absorbing Set is Sublinear Functional | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $A$ be a convex absorbing set.
Let $\mu_A$ be the Minkowski functional of $A$.
Then $\mu_A$ is a sublinear functional. | This follows from Minkowski Functional of Convex Absorbing Set is Positive Homogeneous and Minkowski Functional of Convex Absorbing Set is Sublinear.
{{qed}}
Category:Minkowski Functionals
Category:Sublinear Functionals
7m4ybn4w9pnm8v4lmuvohqpwfjg21y2 | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $A$ be a [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Absorbing Set|absorbing set]].
Let $\mu_A$ be the [[Definition:Minkowski Functional|Minkowski functional]] of $A$.
Then $\mu_A$ is a [[Definiti... | This follows from [[Minkowski Functional of Convex Absorbing Set is Positive Homogeneous]] and [[Minkowski Functional of Convex Absorbing Set is Sublinear]].
{{qed}}
[[Category:Minkowski Functionals]]
[[Category:Sublinear Functionals]]
7m4ybn4w9pnm8v4lmuvohqpwfjg21y2 | Minkowski Functional of Convex Absorbing Set is Sublinear Functional | https://proofwiki.org/wiki/Minkowski_Functional_of_Convex_Absorbing_Set_is_Sublinear_Functional | https://proofwiki.org/wiki/Minkowski_Functional_of_Convex_Absorbing_Set_is_Sublinear_Functional | [
"Minkowski Functionals",
"Sublinear Functionals"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Absorbing Set",
"Definition:Minkowski Functional",
"Definition:Sublinear Functional"
] | [
"Minkowski Functional of Convex Absorbing Set is Positive Homogeneous",
"Minkowski Functional of Convex Absorbing Set is Sublinear",
"Category:Minkowski Functionals",
"Category:Sublinear Functionals"
] |
proofwiki-20933 | Open Convex Set in Hausdorff Locally Convex Space is Separated from Points outside Set by Continuous Linear Functional | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\GF$ equipped with its standard topology.
Let $X^\ast$ be the topological dual of $X$.
Let $C$ be an open convex set such that ${\mathbf 0}_X \in C$.
Let $x_0 \in X \setminus C$.
Then there exists $f \in X^\ast$ such that: ... | === Case 1: $\GF = \R$ ===
Let $\mu_C$ be the Minkowski functional of $C$.
From Minkowski Functional of Convex Absorbing Set is Sublinear Functional, $\mu_A$ is a sublinear functional.
Let $Y = \span \set {x_0}$.
Define $g : Y \to \R$ by:
:$\map g {\lambda x_0} = \lambda \map {\mu_C} {x_0}$
for each $\lambda \in \R$... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Hausdorff Locally Convex Space|Hausdorff locally convex space]] over $\GF$ equipped with its [[Definition:Locally Convex Space/Standard Topology|standard topology]].
Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual]] of $X$... | === Case 1: $\GF = \R$ ===
Let $\mu_C$ be the [[Definition:Minkowski Functional of Convex Absorbing Set|Minkowski functional]] of $C$.
From [[Minkowski Functional of Convex Absorbing Set is Sublinear Functional]], $\mu_A$ is a [[Definition:Sublinear Functional|sublinear functional]].
Let $Y = \span \set {x_0}$.
D... | Open Convex Set in Hausdorff Locally Convex Space is Separated from Points outside Set by Continuous Linear Functional | https://proofwiki.org/wiki/Open_Convex_Set_in_Hausdorff_Locally_Convex_Space_is_Separated_from_Points_outside_Set_by_Continuous_Linear_Functional | https://proofwiki.org/wiki/Open_Convex_Set_in_Hausdorff_Locally_Convex_Space_is_Separated_from_Points_outside_Set_by_Continuous_Linear_Functional | [
"Topological Dual Spaces",
"Locally Convex Spaces",
"Topological Dual Spaces"
] | [
"Definition:Locally Convex Space/Hausdorff",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Topological Dual Space",
"Definition:Open Set",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Minkowski Functional",
"Minkowski Functional of Convex Absorbing Set is Sublinear Functional",
"Definition:Sublinear Functional",
"Hahn-Banach Theorem/Real Vector Space",
"Definition:Linear Functional",
"Definition:Extension of Mapping",
"Definition:Continuous Mapping",
"Definition:Open Ne... |
proofwiki-20934 | Dual Operator is Well-Defined | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ be a bounded linear transformation.
Let $X^\ast$ and $Y^\ast$ be the normed duals of $X$ and $Y$ respectively.
Then the dual operator $T : Y^\ast \to X^\ast$ is well-defined. | We first show that $T^\ast$ is well-defined as a mapping $Y^\ast \to X^\ast$.
That is, we want to show that $T^\ast f \in X^\ast$ for each $f \in Y^\ast$.
For $x, y \in X$ and $\lambda, \mu \in \GF$, we have:
{{begin-eqn}}
{{eqn | l = \map {\paren {T^\ast f} } {\alpha x + \beta y}
| r = \map f {\map T {\alpha x + \... | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Let $X^\ast$ and $Y^\ast$ be the [[Definition:Normed Dual Space|normed duals]] of $X$ and $Y$ respect... | We first show that $T^\ast$ is well-defined as a [[Definition:Mapping|mapping]] $Y^\ast \to X^\ast$.
That is, we want to show that $T^\ast f \in X^\ast$ for each $f \in Y^\ast$.
For $x, y \in X$ and $\lambda, \mu \in \GF$, we have:
{{begin-eqn}}
{{eqn | l = \map {\paren {T^\ast f} } {\alpha x + \beta y}
| r = \m... | Dual Operator is Well-Defined | https://proofwiki.org/wiki/Dual_Operator_is_Well-Defined | https://proofwiki.org/wiki/Dual_Operator_is_Well-Defined | [
"Dual Operators"
] | [
"Definition:Normed Vector Space",
"Definition:Bounded Linear Transformation",
"Definition:Normed Dual Space",
"Definition:Dual Operator"
] | [
"Definition:Mapping",
"Definition:Linear Functional",
"Definition:Bounded Linear Functional",
"Fundamental Property of Norm on Bounded Linear Functional",
"Fundamental Property of Norm on Bounded Linear Transformation",
"Definition:Bounded Linear Functional"
] |
proofwiki-20935 | Dual Operator is Bounded Linear Transformation | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ be a bounded linear transformation.
Let $X^\ast$ and $Y^\ast$ be the normed duals of $X$ and $Y$ respectively.
Then the dual operator $T^\ast : Y^\ast \to X^\ast$ is a bounded linear transformation.
Further:
:$\norm {T^... | Let $f, g \in Y^\ast$ and $\alpha, \beta \in \GF$.
Then we have, for each $x \in X$:
{{begin-eqn}}
{{eqn | l = \map {\paren {\map {T^\ast} {\alpha f + \beta g} } } x
| r = \map {\paren {\alpha f + \beta g} } {T x}
}}
{{eqn | r = \alpha \map f {T x} + \beta \map g {T x}
}}
{{eqn | r = \alpha \map {\paren {T^\ast f} }... | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Let $X^\ast$ and $Y^\ast$ be the [[Definition:Normed Dual Space|normed duals]] of $X$ and $Y$ respect... | Let $f, g \in Y^\ast$ and $\alpha, \beta \in \GF$.
Then we have, for each $x \in X$:
{{begin-eqn}}
{{eqn | l = \map {\paren {\map {T^\ast} {\alpha f + \beta g} } } x
| r = \map {\paren {\alpha f + \beta g} } {T x}
}}
{{eqn | r = \alpha \map f {T x} + \beta \map g {T x}
}}
{{eqn | r = \alpha \map {\paren {T^\ast f}... | Dual Operator is Bounded Linear Transformation | https://proofwiki.org/wiki/Dual_Operator_is_Bounded_Linear_Transformation | https://proofwiki.org/wiki/Dual_Operator_is_Bounded_Linear_Transformation | [
"Dual Operators",
"Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Bounded Linear Transformation",
"Definition:Normed Dual Space",
"Definition:Dual Operator",
"Definition:Bounded Linear Transformation"
] | [
"Dual Operator is Well-Defined",
"Definition:Sequence",
"Definition:Support Functional"
] |
proofwiki-20936 | Normed Vector Space is Separable iff Weakly Separable | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.
Let $w$ be the weak topology on $X$.
Then $\struct {X, \norm {\, \cdot \,} }$ is separable {{iff}} $\struct {X, w}$ is separable. | === Necessary Condition ===
This follows from Separable Topological Space remains Separable in Coarser Topology.
{{qed}} | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $X$.
Then $\struct {X, \norm {\, \cdot \,} }$ is [[Definition:Separable Space|separab... | === Necessary Condition ===
This follows from [[Separable Topological Space remains Separable in Coarser Topology]].
{{qed}} | Normed Vector Space is Separable iff Weakly Separable | https://proofwiki.org/wiki/Normed_Vector_Space_is_Separable_iff_Weakly_Separable | https://proofwiki.org/wiki/Normed_Vector_Space_is_Separable_iff_Weakly_Separable | [
"Weak Topologies on Topological Vector Spaces",
"Separable Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Separable Space",
"Definition:Separable Space"
] | [
"Separable Topological Space remains Separable in Coarser Topology"
] |
proofwiki-20937 | Separable Topological Space remains Separable in Coarser Topology | Let $\struct {X, \tau_2}$ be a separable topological space.
Let $\tau_1$ be a topology on $X$ such that:
:$\tau_1 \subseteq \tau_2$
That is, such that $\tau_1$ is coarser than $\tau_2$.
Then $\struct {X, \tau_1}$ is separable. | Let $\cl_1$ and $\cl_2$ denote the topological closure in $\struct {X, \tau_1}$ and $\struct {X, \tau_2}$ respectively.
Let $D$ be a countable everywhere dense subset of $\struct {X, \tau_2}$.
From Topological Closure in Coarser Topology is Larger, we have:
:$\map {\cl_2} D \subseteq \map {\cl_1} D$
Since $D$ is every... | Let $\struct {X, \tau_2}$ be a [[Definition:Separable Space|separable]] [[Definition:Topological Space|topological space]].
Let $\tau_1$ be a [[Definition:Topology|topology]] on $X$ such that:
:$\tau_1 \subseteq \tau_2$
That is, such that $\tau_1$ is [[Definition:Coarser Topology|coarser]] than $\tau_2$.
Then $\s... | Let $\cl_1$ and $\cl_2$ denote the [[Definition:Topological Closure|topological closure]] in $\struct {X, \tau_1}$ and $\struct {X, \tau_2}$ respectively.
Let $D$ be a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere dense subset]] of $\struct {X, \tau_2}$.
From [[Topological Closure in... | Separable Topological Space remains Separable in Coarser Topology | https://proofwiki.org/wiki/Separable_Topological_Space_remains_Separable_in_Coarser_Topology | https://proofwiki.org/wiki/Separable_Topological_Space_remains_Separable_in_Coarser_Topology | [
"Separable Spaces"
] | [
"Definition:Separable Space",
"Definition:Topological Space",
"Definition:Topology",
"Definition:Coarser Topology",
"Definition:Separable Space"
] | [
"Definition:Closure (Topology)",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Topological Closure in Coarser Topology is Larger",
"Definition:Everywhere Dense",
"Definition:Everywhere Dense",
"Category:Separable Spaces"
] |
proofwiki-20938 | Point in Set Closure iff Limit of Net | Let $\struct {X, \tau}$ be a topological space.
Let $E \subseteq X$ be a subset.
Let $\map \cl E$ be the closure of $E$ in $\struct {X, \tau}$.
Then $x \in \map \cl E$ {{iff}} there exists a directed set $\struct {\Lambda, \preceq}$ and a net $\family {x_\lambda}_{\lambda \in \Lambda}$ in $E$ converging to $x$. | === Necessary Condition ===
Let $x \in \map \cl E$.
Let $\Lambda$ be the set of open neighborhoods of $x$ in $\struct {X, \tau}$.
From Open Neighborhoods of Point form Directed Ordering, $\struct {\Lambda, \supseteq}$ is a directed set.
For each $U \in \Lambda$, we have $U \cap E \ne \O$ from the definition of closure... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $E \subseteq X$ be a [[Definition:Subset|subset]].
Let $\map \cl E$ be the [[Definition:Topological Closure|closure]] of $E$ in $\struct {X, \tau}$.
Then $x \in \map \cl E$ {{iff}} there exists a [[Definition:Directed Set|directed... | === Necessary Condition ===
Let $x \in \map \cl E$.
Let $\Lambda$ be the [[Definition:Set|set]] of [[Definition:Open Neighborhood|open neighborhoods]] of $x$ in $\struct {X, \tau}$.
From [[Open Neighborhoods of Point form Directed Ordering]], $\struct {\Lambda, \supseteq}$ is a [[Definition:Directed Set|directed se... | Point in Set Closure iff Limit of Net | https://proofwiki.org/wiki/Point_in_Set_Closure_iff_Limit_of_Net | https://proofwiki.org/wiki/Point_in_Set_Closure_iff_Limit_of_Net | [
"Moore-Smith Sequences",
"Nets (Set Theory)",
"Nets (Set Theory)",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Closure (Topology)",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net"
] | [
"Definition:Set",
"Definition:Open Neighborhood",
"Open Neighborhoods of Point form Directed Ordering",
"Definition:Directed Preordering",
"Definition:Closure (Topology)",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Open Neighborhood",
"Definition:Convergent Net",
"Defi... |
proofwiki-20939 | Characterization of Openness in terms of Nets | Let $\struct {X, \tau}$ be a topological space.
Let $U \subseteq X$.
Then $U \in \tau$ {{iff}} for each:
:$x \in U$
:directed set $\tuple {\Lambda, \preceq}$
:net $\family {x_\lambda}_{\lambda \in \Lambda}$ converging to $x$
there exists $\lambda \in \Lambda$ with $x_\lambda \in U$. | === Necessary Condition ===
Suppose that $U \in \tau$.
Take $x \in U$, a directed set $\tuple {\Lambda, \preceq}$ and a net $\family {x_\lambda}_{\lambda \in \Lambda}$ converging to $x$.
By the definition of convergence for net, we have:
:there exists $\lambda_0 \in \Lambda$ such that for all $\lambda \in \Lambda$ wi... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $U \subseteq X$.
Then $U \in \tau$ {{iff}} for each:
:$x \in U$
:[[Definition:Directed Set|directed set]] $\tuple {\Lambda, \preceq}$
:[[Definition:Net (Set Theory)|net]] $\family {x_\lambda}_{\lambda \in \Lambda}$ [[Definition:Con... | === Necessary Condition ===
Suppose that $U \in \tau$.
Take $x \in U$, a [[Definition:Directed Set|directed set]] $\tuple {\Lambda, \preceq}$ and a [[Definition:Net (Set Theory)|net]] $\family {x_\lambda}_{\lambda \in \Lambda}$ [[Definition:Convergent Net|converging]] to $x$.
By the definition of [[Definition:Conve... | Characterization of Openness in terms of Nets | https://proofwiki.org/wiki/Characterization_of_Openness_in_terms_of_Nets | https://proofwiki.org/wiki/Characterization_of_Openness_in_terms_of_Nets | [
"Moore-Smith Sequences",
"Nets (Set Theory)",
"Nets (Set Theory)"
] | [
"Definition:Topological Space",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net"
] | [
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Convergent Net",
"Definition:Net (Set Theory)",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Directed Preordering",
"Definition:N... |
proofwiki-20940 | Characterization of Continuity in terms of Nets | Let $\struct {X, \tau_X}$ and $\struct {Y, \tau_Y}$ be topological spaces.
Let $f : \struct {X, \tau_X} \to \struct {Y, \tau_Y}$ be a function.
Let $x \in X$.
Then $f$ is continuous at $x$ {{iff}}:
:for every directed set $\struct {\Lambda, \preceq}$ and net $\family {x_\lambda}_{\lambda \in \Lambda}$ such that $\fami... | === Necessary Condition ===
Suppose that $f$ is continuous at $x$.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {x_\lambda}_{\lambda \in \Lambda}$ be a net such that $\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$.
Let $U$ be an open neighborhood of $\map f x$ in $\struct {Y, \tau_Y}$.
... | Let $\struct {X, \tau_X}$ and $\struct {Y, \tau_Y}$ be [[Definition:Topological Space|topological spaces]].
Let $f : \struct {X, \tau_X} \to \struct {Y, \tau_Y}$ be a [[Definition:Function|function]].
Let $x \in X$.
Then $f$ is [[Definition:Continuous Mapping at Point (Topology)|continuous]] at $x$ {{iff}}:
:for ... | === Necessary Condition ===
Suppose that $f$ is [[Definition:Continuous Mapping at Point (Topology)|continuous]] at $x$.
Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {x_\lambda}_{\lambda \in \Lambda}$ be a [[Definition:Net (Set Theory)|net]] such that $\family {x_\lamb... | Characterization of Continuity in terms of Nets | https://proofwiki.org/wiki/Characterization_of_Continuity_in_terms_of_Nets | https://proofwiki.org/wiki/Characterization_of_Continuity_in_terms_of_Nets | [
"Moore-Smith Sequences",
"Nets (Set Theory)",
"Continuous Mappings",
"Nets (Set Theory)"
] | [
"Definition:Topological Space",
"Definition:Function",
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Net (Set Theory)",
"Definition:Convergent Net"
] | [
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Convergent Net",
"Definition:Open Neighborhood",
"Definition:Convergent Net",
... |
proofwiki-20941 | Existence of Upper Bound of Finite Subset of Directed Set | Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\FF \subseteq \Lambda$ be a finite subset.
Then $\FF$ has an upper bound in $\struct {\Lambda, \preceq}$. | We induct on the cardinality of $\FF$.
Let $n \in \Z_{\ge 0}$.
For all $m \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:if $\FF \subseteq \Lambda$ has $\card \FF = n$, then $\FF$ has an upper bound. | Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\FF \subseteq \Lambda$ be a [[Definition:Finite Subset|finite subset]].
Then $\FF$ has an [[Definition:Upper Bound of Subset of Preordered Set|upper bound]] in $\struct {\Lambda, \preceq}$. | We induct on the [[Definition:Cardinality|cardinality]] of $\FF$.
Let $n \in \Z_{\ge 0}$.
For all $m \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:if $\FF \subseteq \Lambda$ has $\card \FF = n$, then $\FF$ has an [[Definition:Upper Bound of Subset of Preordered Set|upper bound]]. | Existence of Upper Bound of Finite Subset of Directed Set | https://proofwiki.org/wiki/Existence_of_Upper_Bound_of_Finite_Subset_of_Directed_Set | https://proofwiki.org/wiki/Existence_of_Upper_Bound_of_Finite_Subset_of_Directed_Set | [
"Order Theory"
] | [
"Definition:Directed Preordering",
"Definition:Finite Subset",
"Definition:Upper Bound of Subset of Preordered Set"
] | [
"Definition:Cardinality",
"Definition:Proposition",
"Definition:Upper Bound of Subset of Preordered Set",
"Definition:Upper Bound of Subset of Preordered Set",
"Definition:Upper Bound of Subset of Preordered Set",
"Definition:Upper Bound of Subset of Preordered Set"
] |
proofwiki-20942 | Characterization of Convergent Net in Weak Topology | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$ admitting a weak topology $w$.
Let $X^\ast$ be the topological dual of $X$.
Let $x \in X$.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {x_\lambda}_{\lambda \in \Lambda}$ be a net in $X$.
Then:
:$\fami... | === Necessary Condition ===
Suppose that:
:$\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$ in $\struct {X, w}$.
Let $f \in X^\ast$.
From Characterization of Continuity of Linear Functional in Weak Topology, $f : \struct {X, w} \to \GF$ is continuous.
From Characterization of Continuity in terms of Nets, ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$ admitting a [[Definition:Weak Topology on Topological Vector Space|weak topology]] $w$.
Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual]] of $X$.
Let $x \i... | === Necessary Condition ===
Suppose that:
:$\family {x_\lambda}_{\lambda \in \Lambda}$ [[Definition:Convergent Net|converges]] to $x$ in $\struct {X, w}$.
Let $f \in X^\ast$.
From [[Characterization of Continuity of Linear Functional in Weak Topology]], $f : \struct {X, w} \to \GF$ is [[Definition:Continuous Mapp... | Characterization of Convergent Net in Weak Topology | https://proofwiki.org/wiki/Characterization_of_Convergent_Net_in_Weak_Topology | https://proofwiki.org/wiki/Characterization_of_Convergent_Net_in_Weak_Topology | [
"Moore-Smith Sequences",
"Nets (Set Theory)",
"Weak Topologies on Topological Vector Spaces",
"Nets (Set Theory)"
] | [
"Definition:Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Topological Dual Space",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Net (Set Theory)",
"Definition:Convergent Net"
] | [
"Definition:Convergent Net",
"Characterization of Continuity of Linear Functional in Weak Topology",
"Definition:Continuous Mapping",
"Characterization of Continuity in terms of Nets",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Net (Set Theory)",
"Definition:Convergent Net... |
proofwiki-20943 | Characterization of Convergent Net in Weak-* Topology | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space.
Let $X^\ast$ be the topological dual of $X$.
Let $w^\ast$ be the weak-$\ast$ topology on $X$.
Let $f \in X^\ast$.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {f_\lambda}_{\lambda \in \Lambda}$ be a net.
Then:
:$... | === Necessary Condition ===
Suppose that:
:$\family {f_\lambda}_{\lambda \in \Lambda}$ converges to $f$ in $\struct {X^\ast, w^\ast}$
Let $x \in X$.
From Characterization of Continuity of Linear Functional in Weak-* Topology, $x^\wedge : \struct {X^\ast, w^\ast} \to \GF$ is continuous.
From Characterization of Contin... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]].
Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual]] of $X$.
Let $w^\ast$ be the [[Definition:Weak-* Topology|weak-$\ast$ topology]] on $X$.
Let $f \in X^\ast$.
Let $... | === Necessary Condition ===
Suppose that:
:$\family {f_\lambda}_{\lambda \in \Lambda}$ [[Definition:Convergent Net|converges]] to $f$ in $\struct {X^\ast, w^\ast}$
Let $x \in X$.
From [[Characterization of Continuity of Linear Functional in Weak-* Topology]], $x^\wedge : \struct {X^\ast, w^\ast} \to \GF$ is [[Defi... | Characterization of Convergent Net in Weak-* Topology | https://proofwiki.org/wiki/Characterization_of_Convergent_Net_in_Weak-*_Topology | https://proofwiki.org/wiki/Characterization_of_Convergent_Net_in_Weak-*_Topology | [
"Moore-Smith Sequences",
"Nets (Set Theory)",
"Nets (Set Theory)",
"Weak-* Topologies"
] | [
"Definition:Topological Vector Space",
"Definition:Topological Dual Space",
"Definition:Weak-* Topology",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Net (Set Theory)",
"Definition:Convergent Net"
] | [
"Definition:Convergent Net",
"Characterization of Continuity of Linear Functional in Weak-* Topology",
"Definition:Continuous Mapping",
"Characterization of Continuity in terms of Nets",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:Net (Set Theory)",
"Definition:Convergent N... |
proofwiki-20944 | Characterization of Hausdorff Property in terms of Nets | Let $\struct {X, \tau}$ be a topological space.
We have that $\struct {X, \tau}$ is Hausdorff {{iff}} convergent nets have a unique limit.
That is, {{iff}} whenever $\struct {\Lambda, \preceq}$ is a directed set, $x, y \in X$ and $\family {x_\lambda}_{\lambda \in \Lambda}$ is a net converging to $x$ and $y$ in $\struc... | === Necessary Condition ===
Suppose that $\struct {X, \tau}$ is Hausdorff.
Let $\struct {\Lambda, \preceq}$ is a directed set, $x, y \in X$ and $\family {x_\lambda}_{\lambda \in \Lambda}$ is a net converging to $x$ and $y$ in $\struct {X, \tau}$.
{{AimForCont}} suppose that $x \ne y$.
Since $\struct {X, \tau}$ is Haus... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
We have that $\struct {X, \tau}$ is [[Definition:Hausdorff Space|Hausdorff]] {{iff}} [[Definition:Convergent Net|convergent nets]] have a unique [[Definition:Limit of Net|limit]].
That is, {{iff}} whenever $\struct {\Lambda, \preceq}$ ... | === Necessary Condition ===
Suppose that $\struct {X, \tau}$ is [[Definition:Hausdorff Space|Hausdorff]].
Let $\struct {\Lambda, \preceq}$ is a [[Definition:Directed Set|directed set]], $x, y \in X$ and $\family {x_\lambda}_{\lambda \in \Lambda}$ is a [[Definition:Net (Set Theory)|net]] [[Definition:Convergent Net|co... | Characterization of Hausdorff Property in terms of Nets | https://proofwiki.org/wiki/Characterization_of_Hausdorff_Property_in_terms_of_Nets | https://proofwiki.org/wiki/Characterization_of_Hausdorff_Property_in_terms_of_Nets | [
"Moore-Smith Sequences",
"Nets (Set Theory)",
"Hausdorff Spaces",
"Nets (Set Theory)"
] | [
"Definition:Topological Space",
"Definition:T2 Space",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net"
] | [
"Definition:T2 Space",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Definition:T2 Space",
"Definition:Open Neighborhood",
"Definition:Convergent Net",
"Definition:Convergent Net",
"Definition:Directed Preordering",
"Definition:T2 Space",
"Defini... |
proofwiki-20945 | Characterization of Closedness in terms of Nets | Let $\struct {X, \tau}$ be a topological space.
Let $E \subseteq X$ be a subset.
Then $E$ is closed {{iff}} for each $x \in X$ we have $x \in E$ {{iff}}:
:there exists a directed set $\struct {\Lambda, \preceq}$ and a net $\family {x_\lambda}_{\lambda \in \Lambda}$ in $E$ converging to $x$. | From Set is Closed iff Equals Topological Closure, we have that $E$ is closed {{iff}}:
:$x \in E$ {{iff}} $x \in \map \cl E$
From Point in Set Closure iff Limit of Net, for $x \in X$ we have $x \in \map \cl E$ {{iff}}:
:there exists a directed set $\struct {\Lambda, \preceq}$ and a net $\family {x_\lambda}_{\lambda \in... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $E \subseteq X$ be a [[Definition:Subset|subset]].
Then $E$ is [[Definition:Closed Set|closed]] {{iff}} for each $x \in X$ we have $x \in E$ {{iff}}:
:there exists a [[Definition:Directed Set|directed set]] $\struct {\Lambda, \prec... | From [[Set is Closed iff Equals Topological Closure]], we have that $E$ is [[Definition:Closed Set|closed]] {{iff}}:
:$x \in E$ {{iff}} $x \in \map \cl E$
From [[Point in Set Closure iff Limit of Net]], for $x \in X$ we have $x \in \map \cl E$ {{iff}}:
:there exists a [[Definition:Directed Set|directed set]] $\struct ... | Characterization of Closedness in terms of Nets | https://proofwiki.org/wiki/Characterization_of_Closedness_in_terms_of_Nets | https://proofwiki.org/wiki/Characterization_of_Closedness_in_terms_of_Nets | [
"Moore-Smith Sequences",
"Nets (Set Theory)",
"Nets (Set Theory)"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Closed Set",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net"
] | [
"Set is Closed iff Equals Topological Closure",
"Definition:Closed Set",
"Point in Set Closure iff Limit of Net",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Category:Nets (Set Theory)"
] |
proofwiki-20946 | Dual Operator is Weak-* to Weak-* Continuous | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ be a bounded linear transformation.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ and $\struct {Y^\ast, \norm {\, \cdot \,}_{Y^\ast} }$ be the normed duals of $X$ and $Y$ respectively.
Let $T^\ast : Y^\ast \to X^... | From Continuity in Initial Topology, it is enough to show that for each $\Phi \in \struct {X^\ast, w^\ast}^\ast$, we have:
:$\Phi \circ T^\ast : \struct {Y^\ast, w^\ast} \to \GF$ is Continuous.
From Characterization of Continuity of Linear Functional in Weak-* Topology, we have $\Phi = x^\wedge$ for some $x \in X$, wh... | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ and $\struct {Y^\ast, \norm {\, \cdot \,}_{Y^\a... | From [[Continuity in Initial Topology]], it is enough to show that for each $\Phi \in \struct {X^\ast, w^\ast}^\ast$, we have:
:$\Phi \circ T^\ast : \struct {Y^\ast, w^\ast} \to \GF$ is [[Definition:Continuous Mapping|Continuous]].
From [[Characterization of Continuity of Linear Functional in Weak-* Topology]], we h... | Dual Operator is Weak-* to Weak-* Continuous | https://proofwiki.org/wiki/Dual_Operator_is_Weak-*_to_Weak-*_Continuous | https://proofwiki.org/wiki/Dual_Operator_is_Weak-*_to_Weak-*_Continuous | [
"Dual Operators",
"Weak-* Topologies"
] | [
"Definition:Normed Vector Space",
"Definition:Bounded Linear Transformation",
"Definition:Normed Dual Space",
"Definition:Dual Operator",
"Definition:Continuous Mapping"
] | [
"Continuity in Initial Topology",
"Definition:Continuous Mapping",
"Characterization of Continuity of Linear Functional in Weak-* Topology",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Characterization of Continuity of Linear Functional in Weak-* Topology ",
"Definition:Continuous ... |
proofwiki-20947 | Second Dual Operator Extends Operator | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ be a bounded linear transformation.
Let $X^{\ast \ast}$ and $Y^{\ast \ast}$ be the second normed duals of $X$ and $Y$ respectively.
Let $\iota_X : X \to X^{\ast \ast}$ be the evaluation linear transformation on $X$.
Let... | It is shown in Dual Operator is Weak-* to Weak-* Continuous that for each $x \in X$ we have:
:$x^\wedge \circ T^\ast = \paren {T x}^\wedge$
where $x^\wedge = \iota_X x$ and $\paren {T x}^\wedge = \map {\iota_Y} {T x}$.
From the definition of the second dual operator and the evaluation linear transformations, this is ... | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Let $X^{\ast \ast}$ and $Y^{\ast \ast}$ be the [[Definition:Second Normed Dual|second normed duals]] ... | It is shown in [[Dual Operator is Weak-* to Weak-* Continuous]] that for each $x \in X$ we have:
:$x^\wedge \circ T^\ast = \paren {T x}^\wedge$
where $x^\wedge = \iota_X x$ and $\paren {T x}^\wedge = \map {\iota_Y} {T x}$.
From the definition of the [[Definition:Second Dual Operator|second dual operator]] and the ... | Second Dual Operator Extends Operator | https://proofwiki.org/wiki/Second_Dual_Operator_Extends_Operator | https://proofwiki.org/wiki/Second_Dual_Operator_Extends_Operator | [
"Second Dual Operators"
] | [
"Definition:Normed Vector Space",
"Definition:Bounded Linear Transformation",
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Definition:Second Dual Operator",
"Definition:Extension of... | [
"Dual Operator is Weak-* to Weak-* Continuous",
"Definition:Second Dual Operator",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Category:Second Dual Operators"
] |
proofwiki-20948 | Characterization of Dual Operator | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be normed vector spaces over $\GF$.
Let $X^\ast$ and $Y^\ast$ be the normed dual spaces of $X$ and $Y$ respectively.
Let $X^{\ast \ast}$ and $Y^{\ast \ast}$ be the second normed duals of $X$ and $Y$ respectively.
Let $T : Y^\ast \to X^\ast$ be a bounded linear transformati... | === $(1)$ implies $(2)$ ===
Suppose that $T$ is $\struct {w^\ast, w^\ast}$-continuous.
From Characterization of Continuity of Linear Functional in Weak-* Topology , we have that:
:$x^\wedge : \struct {X^\ast, w^\ast} \to \GF$ is continuous.
From Composite of Continuous Mappings is Continuous:
:$x^\wedge \circ T : \str... | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $X^\ast$ and $Y^\ast$ be the [[Definition:Normed Dual Space|normed dual spaces]] of $X$ and $Y$ respectively.
Let $X^{\ast \ast}$ and $Y^{\ast \ast}$ be the [[Definition:Second Normed Dual|second ... | === $(1)$ implies $(2)$ ===
Suppose that $T$ is [[Definition:Continuous Mapping|$\struct {w^\ast, w^\ast}$-continuous]].
From [[Characterization of Continuity of Linear Functional in Weak-* Topology ]], we have that:
:$x^\wedge : \struct {X^\ast, w^\ast} \to \GF$ is [[Definition:Continuous Mapping|continuous]].
Fr... | Characterization of Dual Operator | https://proofwiki.org/wiki/Characterization_of_Dual_Operator | https://proofwiki.org/wiki/Characterization_of_Dual_Operator | [
"Dual Operators",
"Weak-* Topologies"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Second Normed Dual",
"Definition:Bounded Linear Transformation",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Definition:Continuous Mapp... | [
"Definition:Continuous Mapping",
"Characterization of Continuity of Linear Functional in Weak-* Topology ",
"Definition:Continuous Mapping",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Mapping",
"Characterization of Continuity of Linear Functional in Weak-* Topology ",
"Defi... |
proofwiki-20949 | Characterization of Norm-to-Weak and Weak-to-Weak Continuity of Linear Transformations | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $w_X$ be the weak topology on $X$.
Let $w_Y$ be the weak topology on $Y$.
Let $T : X \to Y$ be a linear transformation.
{{TFAE}}
:$(1): \quad$ $T$ is $\struct {\norm {... | Let $X^\ast$ and $Y^\ast$ be the normed dual spaces of $X$ and $Y$ respectively. | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $w_X$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $X$.
Let $w_Y$ be the [[Definition:Weak Top... | Let $X^\ast$ and $Y^\ast$ be the [[Definition:Normed Dual Space|normed dual spaces]] of $X$ and $Y$ respectively. | Characterization of Norm-to-Weak and Weak-to-Weak Continuity of Linear Transformations | https://proofwiki.org/wiki/Characterization_of_Norm-to-Weak_and_Weak-to-Weak_Continuity_of_Linear_Transformations | https://proofwiki.org/wiki/Characterization_of_Norm-to-Weak_and_Weak-to-Weak_Continuity_of_Linear_Transformations | [
"Linear Transformations",
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Linear Transformation",
"Definition:Continuous Mapping",
"Definition:Continuous Mapping",
"Definition:Continuous Mapping"
] | [
"Definition:Normed Dual Space"
] |
proofwiki-20950 | Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Real 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:
:$A \subseteq \set {x \in X : \map f x < c}$
and:
:$B \subseteq \set {x \in X : \map f x \ge c}$
That is:
:there exists $f \in X^\ast$ and $c \in \R$ such that ... | Fix $a_0 \in A$ and $b_0 \in B$.
From Sum of Set and Open Set in Topological Vector Space is Open, we have:
:$A - B$ is open.
From Translation of Open Set in Topological Vector Space is Open, we have:
:$A - B + \paren {b_0 - a_0}$ is open.
Note that since $A \cap B = \O$, we have $0 \notin A - B$.
Hence $b_0 - a_0... | Let $A \subseteq X$ be an [[Definition:Open Set|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:
:$A \subseteq \set {x ... | Fix $a_0 \in A$ and $b_0 \in B$.
From [[Sum of Set and Open Set in Topological Vector Space is Open]], we have:
:$A - B$ is [[Definition:Open Set|open]].
From [[Translation of Open Set in Topological Vector Space is Open]], we have:
:$A - B + \paren {b_0 - a_0}$ is [[Definition:Open Set|open]].
Note that since... | Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Real Case/Open Convex Set and Convex Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Hausdorff_Locally_Convex_Space/Real_Case/Open_Convex_Set_and_Convex_Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Hausdorff_Locally_Convex_Space/Real_Case/Open_Convex_Set_and_Convex_Set | [
"Hahn-Banach Separation Theorem"
] | [
"Definition:Open Set",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Disjoint Sets"
] | [
"Sum of Set and Open Set in Topological Vector Space is Open",
"Definition:Open Set",
"Translation of Open Set in Topological Vector Space is Open",
"Definition:Open Set",
"Definition:Convex Set (Vector Space)",
"Translation of Convex Set in Vector Space is Convex",
"Definition:Convex Set (Vector Space)... |
proofwiki-20951 | Minkowski Functional of Symmetric Convex Absorbing Set in Real Vector Space is Seminorm | Let $X$ be a vector space over $\R$.
Let $A \subseteq X$ be a set that is symmetric, convex and absorbing.
Let $\mu_A$ be the Minkowski functional of $A$.
Then $\mu_A$ is a seminorm. | From Minkowski Functional of Convex Absorbing Set is Positive Homogeneous, $\mu_A$ is a sublinear functional.
Hence we have:
:$\map {\mu_A} {x + y} \le \map {\mu_A} x + \map {\mu_A} y$ for all $x, y \in X$
and hence {{SeminormAxiom|3}}.
We also have:
:$\map {\mu_A} {r x} = r \map {\mu_A} x$ for all $r \ge 0$ and $x \... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\R$.
Let $A \subseteq X$ be a [[Definition:Set|set]] that is [[Definition:Symmetric Set|symmetric]], [[Definition:Convex Set (Vector Space)|convex]] and [[Definition:Absorbing Set|absorbing]].
Let $\mu_A$ be the [[Definition:Minkowski Functional|Minkowski fu... | From [[Minkowski Functional of Convex Absorbing Set is Positive Homogeneous]], $\mu_A$ is a [[Definition:Sublinear Functional|sublinear functional]].
Hence we have:
:$\map {\mu_A} {x + y} \le \map {\mu_A} x + \map {\mu_A} y$ for all $x, y \in X$
and hence {{SeminormAxiom|3}}.
We also have:
:$\map {\mu_A} {r x} =... | Minkowski Functional of Symmetric Convex Absorbing Set in Real Vector Space is Seminorm | https://proofwiki.org/wiki/Minkowski_Functional_of_Symmetric_Convex_Absorbing_Set_in_Real_Vector_Space_is_Seminorm | https://proofwiki.org/wiki/Minkowski_Functional_of_Symmetric_Convex_Absorbing_Set_in_Real_Vector_Space_is_Seminorm | [
"Minkowski Functionals",
"Seminorms"
] | [
"Definition:Vector Space",
"Definition:Set",
"Definition:Symmetric Set",
"Definition:Convex Set (Vector Space)",
"Definition:Absorbing Set",
"Definition:Minkowski Functional",
"Definition:Seminorm"
] | [
"Minkowski Functional of Convex Absorbing Set is Positive Homogeneous",
"Definition:Sublinear Functional",
"Definition:Symmetric Set",
"Category:Minkowski Functionals",
"Category:Seminorms"
] |
proofwiki-20952 | Intersection of Balanced Sets in Vector Space is Balanced | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a vector space over $\Bbb F$.
Let $\sequence {E_\alpha}_{\alpha \mathop \in I}$ be an $I$-indexed family of balanced subsets of $X$.
Then:
:$\ds E = \bigcap_{\alpha \mathop \in I} E_\alpha$ is balanced. | Let $x \in E$.
Then $x \in E_\alpha$ for all $\alpha \in I$.
Let $\lambda \in \Bbb F$ have $\cmod \lambda \le 1$.
Since $E_\alpha$ is balanced for each $\alpha \in I$, we have $\lambda x \in E_\alpha$ for each $\alpha \in I$.
So $\lambda x \in E$.
{{qed}}
Category:Balanced Sets
Category:Set Intersection
d0l3abt8mcjjpg... | Let $\Bbb F \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\Bbb F$.
Let $\sequence {E_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|$I$-indexed family]] of [[Definition:Balanced Set|balanced subsets]] of $X$.
Then:
:$\ds E = \bigcap_{\alpha \mathop \in I} E_\al... | Let $x \in E$.
Then $x \in E_\alpha$ for all $\alpha \in I$.
Let $\lambda \in \Bbb F$ have $\cmod \lambda \le 1$.
Since $E_\alpha$ is [[Definition:Balanced Set|balanced]] for each $\alpha \in I$, we have $\lambda x \in E_\alpha$ for each $\alpha \in I$.
So $\lambda x \in E$.
{{qed}}
[[Category:Balanced Sets]]
[[C... | Intersection of Balanced Sets in Vector Space is Balanced | https://proofwiki.org/wiki/Intersection_of_Balanced_Sets_in_Vector_Space_is_Balanced | https://proofwiki.org/wiki/Intersection_of_Balanced_Sets_in_Vector_Space_is_Balanced | [
"Balanced Sets",
"Set Intersection"
] | [
"Definition:Vector Space",
"Definition:Indexing Set/Family",
"Definition:Balanced Set",
"Definition:Balanced Set"
] | [
"Definition:Balanced Set",
"Category:Balanced Sets",
"Category:Set Intersection"
] |
proofwiki-20953 | Balanced Set in Vector Space is Symmetric | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $C \subseteq X$ be a balanced set.
Then $C$ is symmetric. | Since $C$ is balanced, we have:
:$s C \subseteq C$ for all $s \in \C$ with $\cmod s \le 1$.
So in particular, setting $s = -1$:
:$-C \subseteq C$
So $C$ is symmetric.
Category:Symmetric Subsets of Vector Spaces
Category:Balanced Sets
mzh45stgo04j1jjdjhc9k6epfl93nun | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $C \subseteq X$ be a [[Definition:Balanced Set|balanced set]].
Then $C$ is [[Definition:Symmetric Subset of Vector Space|symmetric]]. | Since $C$ is [[Definition:Balanced Set|balanced]], we have:
:$s C \subseteq C$ for all $s \in \C$ with $\cmod s \le 1$.
So in particular, setting $s = -1$:
:$-C \subseteq C$
So $C$ is [[Definition:Symmetric Subset of Vector Space|symmetric]].
[[Category:Symmetric Subsets of Vector Spaces]]
[[Category:Balanced Se... | Balanced Set in Vector Space is Symmetric | https://proofwiki.org/wiki/Balanced_Set_in_Vector_Space_is_Symmetric | https://proofwiki.org/wiki/Balanced_Set_in_Vector_Space_is_Symmetric | [
"Symmetric Subsets of Vector Spaces",
"Balanced Sets"
] | [
"Definition:Vector Space",
"Definition:Balanced Set",
"Definition:Symmetric Set/Vector Space"
] | [
"Definition:Balanced Set",
"Definition:Symmetric Set/Vector Space",
"Category:Symmetric Subsets of Vector Spaces",
"Category:Balanced Sets"
] |
proofwiki-20954 | Minkowski Functional of Balanced Convex Absorbing Set in Vector Space is Seminorm | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $A \subseteq X$ be a set that is balanced, convex and absorbing.
Let $\mu_A$ be the Minkowski functional of $A$.
Then $\mu_A$ is a seminorm. | === Case 1: $\GF = \R$ ===
From Balanced Set in Vector Space is Symmetric, $A$ is symmetric.
Hence by Minkowski Functional of Symmetric Convex Absorbing Set in Real Vector Space is Seminorm, $\mu_A$ is a seminorm in this case.
{{qed|lemma}} | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $A \subseteq X$ be a [[Definition:Set|set]] that is [[Definition:Balanced Set|balanced]], [[Definition:Convex Set (Vector Space)|convex]] and [[Definition:Absorbing Set|absorbing]].
Let $\mu_A$ be the [[Definition:Mi... | === Case 1: $\GF = \R$ ===
From [[Balanced Set in Vector Space is Symmetric]], $A$ is [[Definition:Symmetric Subset of Vector Space|symmetric]].
Hence by [[Minkowski Functional of Symmetric Convex Absorbing Set in Real Vector Space is Seminorm]], $\mu_A$ is a [[Definition:Seminorm|seminorm]] in this case.
{{qed|lemma... | Minkowski Functional of Balanced Convex Absorbing Set in Vector Space is Seminorm | https://proofwiki.org/wiki/Minkowski_Functional_of_Balanced_Convex_Absorbing_Set_in_Vector_Space_is_Seminorm | https://proofwiki.org/wiki/Minkowski_Functional_of_Balanced_Convex_Absorbing_Set_in_Vector_Space_is_Seminorm | [
"Minkowski Functionals",
"Seminorms"
] | [
"Definition:Vector Space",
"Definition:Set",
"Definition:Balanced Set",
"Definition:Convex Set (Vector Space)",
"Definition:Absorbing Set",
"Definition:Minkowski Functional",
"Definition:Seminorm"
] | [
"Balanced Set in Vector Space is Symmetric",
"Definition:Symmetric Set/Vector Space",
"Minkowski Functional of Symmetric Convex Absorbing Set in Real Vector Space is Seminorm",
"Definition:Seminorm"
] |
proofwiki-20955 | Characterization of Paracompactness in T3 Space/Lemma 17 | :$\forall n \in \N_{> 0} : \AA_n$ is a discrete set of subsets. | === Lemma 16 ===
{{:Characterization of Paracompactness in T3 Space/Lemma 16}}
Let $n \in \N_{> 0}$.
Let $x \in X$. | :$\forall n \in \N_{> 0} : \AA_n$ is a [[Definition:Discrete Set of Subsets|discrete set of subsets]]. | === [[Characterization of Paracompactness in T3 Space/Lemma 16|Lemma 16]] ===
{{:Characterization of Paracompactness in T3 Space/Lemma 16}}
Let $n \in \N_{> 0}$.
Let $x \in X$. | Characterization of Paracompactness in T3 Space/Lemma 17 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_17 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_17 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Discrete Set of Subsets"
] | [
"Characterization of Paracompactness in T3 Space/Lemma 16"
] |
proofwiki-20956 | Continuous Mappings into Hausdorff Space coinciding on Everywhere Dense Set coincide | Let $\struct {X, \tau_X}$ be a topological space.
Let $\struct {Y, \tau_Y}$ be a Hausdorff space.
Let $D$ be an everywhere dense subset of $X$.
Let $f : X \to Y$ and $g : X \to Y$ be continuous mappings such that:
:$\map f x = \map g x$ for all $x \in D$.
Then:
:$\map f x = \map g x$ for all $x \in X$. | {{AimForCont}} that there exists $x_0 \in X$ such that $\map f {x_0} \ne \map g {x_0}$.
Since $\struct {Y, \tau_Y}$ is Hausdorff, there exists an open neighborhood $U$ of $\map f {x_0}$ and an open neighborhood $V$ of $\map g {x_0}$ such that:
:$U \cap V = \O$
We then have:
:$x_0 \in f^{-1} \sqbrk U \cap g^{-1} \sqbr... | Let $\struct {X, \tau_X}$ be a [[Definition:Topological Space|topological space]].
Let $\struct {Y, \tau_Y}$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $D$ be an [[Definition:Everywhere Dense|everywhere dense subset]] of $X$.
Let $f : X \to Y$ and $g : X \to Y$ be [[Definition:Continuous Mapping (Top... | {{AimForCont}} that there exists $x_0 \in X$ such that $\map f {x_0} \ne \map g {x_0}$.
Since $\struct {Y, \tau_Y}$ is [[Definition:Hausdorff Space|Hausdorff]], there exists an [[Definition:Open Neighborhood|open neighborhood]] $U$ of $\map f {x_0}$ and an [[Definition:Open Neighborhood|open neighborhood]] $V$ of $\ma... | Continuous Mappings into Hausdorff Space coinciding on Everywhere Dense Set coincide/Proof 1 | https://proofwiki.org/wiki/Continuous_Mappings_into_Hausdorff_Space_coinciding_on_Everywhere_Dense_Set_coincide | https://proofwiki.org/wiki/Continuous_Mappings_into_Hausdorff_Space_coinciding_on_Everywhere_Dense_Set_coincide/Proof_1 | [
"Continuous Mappings into Hausdorff Space coinciding on Everywhere Dense Set coincide",
"Hausdorff Spaces",
"Continuous Mappings (Topology)"
] | [
"Definition:Topological Space",
"Definition:T2 Space",
"Definition:Everywhere Dense",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:T2 Space",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Continuous Mapping (Topology)",
"Definition:Set Intersection",
"Definition:Open Set",
"Definition:Open Set/Topology",
"Definition:Topology",
"Definition:Everywhere Dense",
"Proof by Contradiction"
] |
proofwiki-20957 | Continuous Mappings into Hausdorff Space coinciding on Everywhere Dense Set coincide | Let $\struct {X, \tau_X}$ be a topological space.
Let $\struct {Y, \tau_Y}$ be a Hausdorff space.
Let $D$ be an everywhere dense subset of $X$.
Let $f : X \to Y$ and $g : X \to Y$ be continuous mappings such that:
:$\map f x = \map g x$ for all $x \in D$.
Then:
:$\map f x = \map g x$ for all $x \in X$. | Let $x \in X$.
Since $D$ is everywhere dense, we have that $x \in \map \cl D$, where $\map \cl D$ is the topological closure of $D$.
By Point in Set Closure iff Limit of Net, there exists a directed set $\struct {\Lambda, \preceq}$ and a net $\family {x_\lambda}_{\lambda \in \Lambda}$ in $D$ converging to $x$.
From Ch... | Let $\struct {X, \tau_X}$ be a [[Definition:Topological Space|topological space]].
Let $\struct {Y, \tau_Y}$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $D$ be an [[Definition:Everywhere Dense|everywhere dense subset]] of $X$.
Let $f : X \to Y$ and $g : X \to Y$ be [[Definition:Continuous Mapping (Top... | Let $x \in X$.
Since $D$ is [[Definition:Everywhere Dense|everywhere dense]], we have that $x \in \map \cl D$, where $\map \cl D$ is the [[Definition:Topological Closure|topological closure]] of $D$.
By [[Point in Set Closure iff Limit of Net]], there exists a [[Definition:Directed Set|directed set]] $\struct {\Lamb... | Continuous Mappings into Hausdorff Space coinciding on Everywhere Dense Set coincide/Proof 2 | https://proofwiki.org/wiki/Continuous_Mappings_into_Hausdorff_Space_coinciding_on_Everywhere_Dense_Set_coincide | https://proofwiki.org/wiki/Continuous_Mappings_into_Hausdorff_Space_coinciding_on_Everywhere_Dense_Set_coincide/Proof_2 | [
"Continuous Mappings into Hausdorff Space coinciding on Everywhere Dense Set coincide",
"Hausdorff Spaces",
"Continuous Mappings (Topology)"
] | [
"Definition:Topological Space",
"Definition:T2 Space",
"Definition:Everywhere Dense",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Everywhere Dense",
"Definition:Closure (Topology)",
"Point in Set Closure iff Limit of Net",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Characterization of Continuity in terms of Nets",
"Definition:Net (Set Theory)",
"Definition:Conver... |
proofwiki-20958 | Characterization of Paracompactness in T3 Space/Lemma 18 | :$\AA$ is a cover of $X$ | Let $x \in X$.
We have {{hypothesis}}:
:$\forall n \in \N_{>0} : \Delta_X \subseteq U_n$
By definition of image:
:$\forall n \in \N_{>0} : x \in \map {U_n} x$
Let:
:$y = \min \set{z \in X : x \in \ds \bigcup_{n \in N} \map {U_n} z}$
with respect to the well-ordering $\preccurlyeq$.
By choice of $y$
:$\exists n \in \N_{... | :$\AA$ is a [[Definition:Cover of Set|cover]] of $X$ | Let $x \in X$.
We have {{hypothesis}}:
:$\forall n \in \N_{>0} : \Delta_X \subseteq U_n$
By definition of [[Definition:Image of Element under Relation|image]]:
:$\forall n \in \N_{>0} : x \in \map {U_n} x$
Let:
:$y = \min \set{z \in X : x \in \ds \bigcup_{n \in N} \map {U_n} z}$
with respect to the [[Definition:W... | Characterization of Paracompactness in T3 Space/Lemma 18 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_18 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_18 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Cover of Set"
] | [
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Well-Ordering",
"Definition:Set Union",
"Definition:Set Difference",
"Definition:Cover of Set",
"Category:Characterization of Paracompactness in T3 Space"
] |
proofwiki-20959 | Product of Directed Sets is Directed Set | Let $\struct {\Lambda_1, \preceq_1}$ and $\struct {\Lambda_2, \preceq_2}$ be directed sets.
Define a relation $\sqsubseteq$ on $\Lambda_1 \times \Lambda_2$ by:
:$\tuple {\lambda_1, \lambda_2} \sqsubseteq \tuple {\mu_1, \mu_2}$
for $\tuple {\lambda_1, \lambda_2}, \tuple {\mu_1, \mu_2} \in \Lambda_1 \times \Lambda_2$ {... | From Product of Preordered Sets is Preordered, $\sqsubseteq$ is a preordering.
We now show that $\sqsubseteq$ is directed.
Let $\tuple {\lambda_1, \mu_1}, \tuple {\lambda_2, \mu_2} \in \Lambda_1 \times \Lambda_2$.
Then since $\preceq_1$ is directed, there exists $\lambda \in \Lambda_1$ such that:
:$\lambda_1 \preceq_... | Let $\struct {\Lambda_1, \preceq_1}$ and $\struct {\Lambda_2, \preceq_2}$ be [[Definition:Directed Set|directed sets]].
Define a [[Definition:Relation|relation]] $\sqsubseteq$ on $\Lambda_1 \times \Lambda_2$ by:
:$\tuple {\lambda_1, \lambda_2} \sqsubseteq \tuple {\mu_1, \mu_2}$
for $\tuple {\lambda_1, \lambda_2}, ... | From [[Product of Preordered Sets is Preordered]], $\sqsubseteq$ is a [[Definition:Preordering|preordering]].
We now show that $\sqsubseteq$ is [[Definition:Directed Set|directed]].
Let $\tuple {\lambda_1, \mu_1}, \tuple {\lambda_2, \mu_2} \in \Lambda_1 \times \Lambda_2$.
Then since $\preceq_1$ is [[Definition:Dire... | Product of Directed Sets is Directed Set | https://proofwiki.org/wiki/Product_of_Directed_Sets_is_Directed_Set | https://proofwiki.org/wiki/Product_of_Directed_Sets_is_Directed_Set | [
"Directed Preorderings"
] | [
"Definition:Directed Preordering",
"Definition:Relation",
"Definition:Directed Preordering"
] | [
"Product of Preordered Sets is Preordered",
"Definition:Preordering",
"Definition:Directed Preordering",
"Definition:Directed Preordering",
"Definition:Directed Preordering",
"Definition:Directed Preordering",
"Category:Directed Preorderings"
] |
proofwiki-20960 | Characterization of Convergent Net in Metric Space | Let $\struct {X, d}$ be a metric space.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a net in $X$.
Let $x \in X$.
Then $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ converges in $\struct {X, d}$ {{iff}}:
:for each $\epsilon > 0$ there exists $\la... | === Necessary Condition ===
Suppose that $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ converges to $x$ in $\struct {X, d}$.
Let $\epsilon > 0$.
From Open Ball is Open Set, the open ball $\map {B_\epsilon} x$ with radius $\epsilon$ and center $x$ is open in $\struct {X, d}$.
Since $\family {x_\lambda}_{\lambda \... | Let $\struct {X, d}$ be a [[Definition:Metric Space|metric space]].
Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a [[Definition:Net (Set Theory)|net]] in $X$.
Let $x \in X$.
Then $\family {x_\lambda}_{\lambda \mathop \i... | === Necessary Condition ===
Suppose that $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ [[Definition:Convergent Net|converges]] to $x$ in $\struct {X, d}$.
Let $\epsilon > 0$.
From [[Open Ball is Open Set]], the [[Definition:Open Ball|open ball]] $\map {B_\epsilon} x$ with [[Definition:Radius of Open Ball|radi... | Characterization of Convergent Net in Metric Space | https://proofwiki.org/wiki/Characterization_of_Convergent_Net_in_Metric_Space | https://proofwiki.org/wiki/Characterization_of_Convergent_Net_in_Metric_Space | [
"Metric Spaces",
"Nets (Set Theory)"
] | [
"Definition:Metric Space",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net"
] | [
"Definition:Convergent Net",
"Open Ball is Open Set",
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Definition:Open Set/Metric Space",
"Definition:Convergent Net",
"Definition:Open Set/Metric Space"
] |
proofwiki-20961 | Characterization of Even Cover | Let $T = \struct {S, \tau}$ be a topological space.
Let $\UU$ be a cover of $S$.
Let $\tau_{S \times S}$ denote the product topology on the cartesian product $S \times S$.
Then $\UU$ is an even cover of $T$ {{iff}} there exists an open neighborhood $V$ of the diagonal $\Delta_S$ of $S \times S$ in the product space $\s... | === Necessary Condition ===
Let $\UU$ be an even cover of $T$.
By definition of even cover, there exists a neighborhood $N$ of the diagonal $\Delta_S$ of $S \times S$ in the product space $\struct {S \times S, \tau_{S \times S} }$:
:$\set{\map N x : x \in S}$ is a refinement of $\UU$
By definition of a neighborhood, th... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\UU$ be a [[Definition:Cover of Set|cover]] of $S$.
Let $\tau_{S \times S}$ denote the [[Definition:Product Topology|product topology]] on the [[Definition:Cartesian Product|cartesian product]] $S \times S$.
Then $\UU$ is an ... | === Necessary Condition ===
Let $\UU$ be an [[Definition:Even Cover|even cover]] of $T$.
By definition of [[Definition:Even Cover|even cover]], there exists a [[Definition:Neighborhood of Set|neighborhood]] $N$ of the [[Definition:Diagonal Relation|diagonal $\Delta_S$]] of $S \times S$ in the [[Definition:Product Spa... | Characterization of Even Cover | https://proofwiki.org/wiki/Characterization_of_Even_Cover | https://proofwiki.org/wiki/Characterization_of_Even_Cover | [
"Even Covers"
] | [
"Definition:Topological Space",
"Definition:Cover of Set",
"Definition:Product Topology",
"Definition:Cartesian Product",
"Definition:Even Cover",
"Definition:Open Neighborhood",
"Definition:Diagonal Relation",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Refinement of Cover"... | [
"Definition:Even Cover",
"Definition:Even Cover",
"Definition:Neighborhood (Topology)/Set",
"Definition:Diagonal Relation",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Refinement of Cover",
"Definition:Neighborhood (Topology)/Set",
"Definition:Open Set/Topology",
"Set is Ope... |
proofwiki-20962 | Inverse of Open Set in Product Space is Open in Inverse Product Space | Let $\struct{S_1, \tau_1}$ and $\struct{S_2, \tau_2}$ be topological spaces.
Let $\tau_{S_1 \times S_2}$ be the product topology on the Cartesian product $S_1 \times S_2$.
Let $\tau_{S_2 \times S_1}$ be the product topology on the Cartesian product $S_2 \times S_1$.
Let $V \subseteq S_1 \times S_2$.
Let $V^{-1} \subset... | === Necessary Condition ===
Let $V$ be open in $\struct{S_1 \times S_2, \tau_{S_1 \times S_2}}$.
Let $\tuple{s_2, s_1} \in V^{-1}$
By definition of inverse:
:$\tuple{s_1, s_2} \in V$
By definition of product topology:
:$\BB_{12} = \set{U \times W : U \in \tau_1, W \in \tau_2}$ is a basis for $\tau_{S_1 \times S_2}$
By ... | Let $\struct{S_1, \tau_1}$ and $\struct{S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $\tau_{S_1 \times S_2}$ be the [[Definition:Product Topology|product topology]] on the [[Definition:Cartesian Product|Cartesian product]] $S_1 \times S_2$.
Let $\tau_{S_2 \times S_1}$ be the [[Definition... | === Necessary Condition ===
Let $V$ be [[Definition:Open Set (Topology)|open]] in $\struct{S_1 \times S_2, \tau_{S_1 \times S_2}}$.
Let $\tuple{s_2, s_1} \in V^{-1}$
By definition of [[Definition:Inverse Relation|inverse]]:
:$\tuple{s_1, s_2} \in V$
By definition of [[Definition:Product Topology|product topology... | Inverse of Open Set in Product Space is Open in Inverse Product Space | https://proofwiki.org/wiki/Inverse_of_Open_Set_in_Product_Space_is_Open_in_Inverse_Product_Space | https://proofwiki.org/wiki/Inverse_of_Open_Set_in_Product_Space_is_Open_in_Inverse_Product_Space | [
"Product Spaces"
] | [
"Definition:Topological Space",
"Definition:Product Topology",
"Definition:Cartesian Product",
"Definition:Product Topology",
"Definition:Cartesian Product",
"Definition:Inverse Relation",
"Definition:Relation/Relation as Subset of Cartesian Product",
"Definition:Open Set/Topology",
"Definition:Open... | [
"Definition:Open Set/Topology",
"Definition:Inverse Relation",
"Definition:Product Topology",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Inverse Relation",
"Definition:Product Topology",
"Definition:Basis (Topology)/Analytic Basis",
"Chara... |
proofwiki-20963 | Standard Topology of Locally Convex Space has Local Basis of Balanced Convex Absorbing Sets | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a locally convex space over $\GF$ with standard topology $\tau$.
Then there exists a local basis $\BB$ for ${\mathbf 0}_X$ in $\struct {X, \tau}$ such that:
:each $A \in \BB$ is balanced, convex and absorbing. | For each $\epsilon > 0$ and $S$ a finite subset of $\PP$, set:
:$U_{\epsilon, S} = \set {x \in X : \map p x < \epsilon \text { for each } p \in S}$
and:
:$\BB = \set {U_{\epsilon, S} : \epsilon > 0 \text { and } S \subseteq \PP \text { finite} }$
From Open Sets in Standard Topology of Locally Convex Space, $\BB'$ is a... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Locally Convex Space|locally convex space]] over $\GF$ with [[Definition:Locally Convex Space/Standard Topology|standard topology]] $\tau$.
Then there exists a [[Definition:Local Basis|local basis]] $\BB$ for ${\mathbf 0}_X$ in $\struct {X, \tau... | For each $\epsilon > 0$ and $S$ a [[Definition:Finite Subset|finite subset]] of $\PP$, set:
:$U_{\epsilon, S} = \set {x \in X : \map p x < \epsilon \text { for each } p \in S}$
and:
:$\BB = \set {U_{\epsilon, S} : \epsilon > 0 \text { and } S \subseteq \PP \text { finite} }$
From [[Open Sets in Standard Topology o... | Standard Topology of Locally Convex Space has Local Basis of Balanced Convex Absorbing Sets | https://proofwiki.org/wiki/Standard_Topology_of_Locally_Convex_Space_has_Local_Basis_of_Balanced_Convex_Absorbing_Sets | https://proofwiki.org/wiki/Standard_Topology_of_Locally_Convex_Space_has_Local_Basis_of_Balanced_Convex_Absorbing_Sets | [
"Locally Convex Spaces",
"Balanced Sets",
"Absorbing Sets",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Locally Convex Space",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Local Basis",
"Definition:Balanced Set",
"Definition:Convex Set (Vector Space)",
"Definition:Absorbing Set"
] | [
"Definition:Finite Subset",
"Open Sets in Standard Topology of Locally Convex Space",
"Definition:Local Basis",
"Convex Open Neighborhood of Origin in Topological Vector Space contains Balanced Convex Open Neighborhood",
"Definition:Balanced Set",
"Definition:Convex Set (Vector Space)",
"Definition:Open... |
proofwiki-20964 | Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Real Case/Compact Convex Set and Closed Convex Set | Let $A \subseteq X$ be an compact convex set.
Let $B \subseteq X$ be a closed convex set disjoint from $A$.
Then there exists $f \in X^\ast$ such that:
:$\ds \sup_{x \mathop \in A} \map f x < \inf_{x \mathop \in B} \map f x$ | Since $B$ is closed, $X \setminus B$ is open.
Let $x \in A$.
Since $A \cap B = \O$, we have $x \in X \setminus B$.
So there exists an open neighborhood $O_x$ of $x$ such that $O_x \subseteq X \setminus B$.
From Classification of Open Neighborhoods in Topological Vector Space, there exists an open neighborhood $U_x$ of... | Let $A \subseteq X$ be an [[Definition:Compact Topological Space|compact]] [[Definition:Convex Set (Vector Space)|convex set]].
Let $B \subseteq X$ be a [[Definition:Closed Set|closed]] [[Definition:Convex Set (Vector Space)|convex set]] [[Definition:Disjoint Sets|disjoint]] from $A$.
Then there exists $f \in X^\as... | Since $B$ is [[Definition:Closed Set|closed]], $X \setminus B$ is [[Definition:Open Set|open]].
Let $x \in A$.
Since $A \cap B = \O$, we have $x \in X \setminus B$.
So there exists an [[Definition:Open Neighborhood|open neighborhood]] $O_x$ of $x$ such that $O_x \subseteq X \setminus B$.
From [[Classification of O... | Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Real Case/Compact Convex Set and Closed Convex Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Hausdorff_Locally_Convex_Space/Real_Case/Compact_Convex_Set_and_Closed_Convex_Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Hausdorff_Locally_Convex_Space/Real_Case/Compact_Convex_Set_and_Closed_Convex_Set | [
"Hahn-Banach Separation Theorem"
] | [
"Definition:Compact Topological Space",
"Definition:Convex Set (Vector Space)",
"Definition:Closed Set",
"Definition:Convex Set (Vector Space)",
"Definition:Disjoint Sets"
] | [
"Definition:Closed Set",
"Definition:Open Set",
"Definition:Open Neighborhood",
"Classification of Open Neighborhoods in Topological Vector Space",
"Definition:Open Neighborhood",
"Definition:Symmetric Set",
"Definition:Open Neighborhood",
"Definition:Compact Topological Space",
"Definition:Topology... |
proofwiki-20965 | Realification of Topological Vector Space is Topological Vector Space | Let $\struct {X, \tau}$ be a topological vector space over $\C$.
Let $X_\R$ be the realification of $X$.
Then $\struct {X_\R, \tau}$ is a topological vector space over $\R$.
Further, $\struct {X, \tau}$ is a Hausdorff topological vector space {{iff}} $\struct {X_\R, \tau}$ is a Hausdorff topological vector space. | Since $\struct {X, \tau}$ is a topological vector space, the vector addition map $+_X : X \times X \to X$ is continuous.
Note that the underlying sets and topologies of $X_\R$ and $X$ are identical, we are only restricting the scalar field.
So the vector addition map $+_{X_\R} : X_\R \times X_\R \to X_\R$ is also conti... | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\C$.
Let $X_\R$ be the [[Definition:Realification of Complex Vector Space|realification]] of $X$.
Then $\struct {X_\R, \tau}$ is a [[Definition:Topological Vector Space|topological vector space]] over $\R$.
Further,... | Since $\struct {X, \tau}$ is a [[Definition:Topological Vector Space|topological vector space]], the [[Definition:Vector Addition|vector addition]] [[Definition:Mapping|map]] $+_X : X \times X \to X$ is [[Definition:Continuous Mapping|continuous]].
Note that the [[Definition:Underlying Set|underlying sets]] and [[Defi... | Realification of Topological Vector Space is Topological Vector Space | https://proofwiki.org/wiki/Realification_of_Topological_Vector_Space_is_Topological_Vector_Space | https://proofwiki.org/wiki/Realification_of_Topological_Vector_Space_is_Topological_Vector_Space | [
"Topological Vector Spaces",
"Realifications of Complex Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Realification of Complex Vector Space",
"Definition:Topological Vector Space",
"Definition:Hausdorff Topological Vector Space",
"Definition:Hausdorff Topological Vector Space"
] | [
"Definition:Topological Vector Space",
"Definition:Vector Addition",
"Definition:Mapping",
"Definition:Continuous Mapping",
"Definition:Underlying Set",
"Definition:Topology",
"Definition:Restriction/Mapping",
"Definition:Scalar Field",
"Definition:Vector Addition",
"Definition:Mapping",
"Defini... |
proofwiki-20966 | Hahn-Banach Separation Theorem/Hausdorff Locally Convex 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 $X_\R$ be the realification of $X$.
Applying Hahn-Banach Separation Theorem: Hausdorff Locally Convex Space: Real Case: Open Convex Set and Convex Set, there exists a continuous $\R$-linear functional $g : X \to \R$ and $c \in \R$ such that:
:$\map g a < c \le \map g b$ for each $a \in A$ and $b \in B$.
From Cont... | Let $A \subseteq X$ be an [[Definition:Open Set|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:
:$A \subseteq \set {x ... | Let $X_\R$ be the [[Definition:Realification of Complex Vector Space|realification]] of $X$.
Applying [[Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Real Case/Open Convex Set and Convex Set|Hahn-Banach Separation Theorem: Hausdorff Locally Convex Space: Real Case: Open Convex Set and Convex Set]], th... | Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Complex Case/Open Convex Set and Convex Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Hausdorff_Locally_Convex_Space/Complex_Case/Open_Convex_Set_and_Convex_Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Hausdorff_Locally_Convex_Space/Complex_Case/Open_Convex_Set_and_Convex_Set | [
"Hahn-Banach Separation Theorem"
] | [
"Definition:Open Set",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Disjoint Sets"
] | [
"Definition:Realification of Complex Vector Space",
"Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Real Case/Open Convex Set and Convex Set",
"Definition:Continuous Mapping",
"Definition:Linear Functional",
"Continuous Real Linear Functional on Complex Topological Vector Space is Real Part o... |
proofwiki-20967 | Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Complex Case/Compact Convex Set and Closed Convex Set | Let $A \subseteq X$ be an compact convex set.
Let $B \subseteq X$ be a closed convex set disjoint from $A$.
Then there exists $f \in X^\ast$ such that:
:$\ds \sup_{x \mathop \in A} \map \Re {\map f x} < \inf_{x \mathop \in B} \map \Re {\map f x}$ | Let $X_\R$ be the realification of $X$.
Applying Hahn-Banach Separation Theorem: Hausdorff Locally Convex Space: Real Case: Compact Convex Set and Closed Convex Set, there exists a continuous $\R$-linear functional $g : X \to \R$ and $c \in \R$ such that:
:$\ds \sup_{x \mathop \in A} \map g x < \inf_{x \mathop \in B}... | Let $A \subseteq X$ be an [[Definition:Compact Topological Space|compact]] [[Definition:Convex Set (Vector Space)|convex set]].
Let $B \subseteq X$ be a [[Definition:Closed Set|closed]] [[Definition:Convex Set (Vector Space)|convex set]] [[Definition:Disjoint Sets|disjoint]] from $A$.
Then there exists $f \in X^\as... | Let $X_\R$ be the [[Definition:Realification of Complex Vector Space|realification]] of $X$.
Applying [[Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Real Case/Compact Convex Set and Closed Convex Set|Hahn-Banach Separation Theorem: Hausdorff Locally Convex Space: Real Case: Compact Convex Set and Clo... | Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Complex Case/Compact Convex Set and Closed Convex Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Hausdorff_Locally_Convex_Space/Complex_Case/Compact_Convex_Set_and_Closed_Convex_Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Hausdorff_Locally_Convex_Space/Complex_Case/Compact_Convex_Set_and_Closed_Convex_Set | [
"Hahn-Banach Separation Theorem"
] | [
"Definition:Compact Topological Space",
"Definition:Convex Set (Vector Space)",
"Definition:Closed Set",
"Definition:Convex Set (Vector Space)",
"Definition:Disjoint Sets"
] | [
"Definition:Realification of Complex Vector Space",
"Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Real Case/Compact Convex Set and Closed Convex Set",
"Definition:Continuous Mapping",
"Definition:Linear Functional",
"Continuous Real Linear Functional on Complex Topological Vector Space is R... |
proofwiki-20968 | Goldstine's Theorem | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $X$.
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the second normed dual of $X$.
Let $w^\ast$ be the $w^... | From Closed Unit Ball in Normed Dual Space is Weak-* Closed, $B_{X^{\ast \ast} }^-$ is closed in $\struct {X^{\ast \ast}, w^\ast}$.
In Normed Vector Space is Reflexive iff Closed Unit Ball in Original Space is Mapped to Closed Unit Ball in Second Dual, it is shown that $\iota B_X^- \subseteq B_{X^{\ast \ast} }$.
Hence... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\a... | From [[Closed Unit Ball in Normed Dual Space is Weak-* Closed]], $B_{X^{\ast \ast} }^-$ is [[Definition:Closed Set|closed]] in $\struct {X^{\ast \ast}, w^\ast}$.
In [[Normed Vector Space is Reflexive iff Closed Unit Ball in Original Space is Mapped to Closed Unit Ball in Second Dual]], it is shown that $\iota B_X^- \... | Goldstine's Theorem | https://proofwiki.org/wiki/Goldstine's_Theorem | https://proofwiki.org/wiki/Goldstine's_Theorem | [
"Weak-* Topologies",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Second Normed Dual",
"Definition:Weak-* Topology",
"Definition:Evaluation Linear Transformation",
"Definition:Closed Unit Ball",
"Definition:Closed Unit Ball"
] | [
"Closed Unit Ball in Normed Dual Space is Weak-* Closed",
"Definition:Closed Set",
"Normed Vector Space is Reflexive iff Closed Unit Ball in Original Space is Mapped to Closed Unit Ball in Second Dual",
"Set Closure Preserves Set Inclusion",
"Set is Closed iff Equals Topological Closure",
"Set Complement ... |
proofwiki-20969 | Spanning Criterion of Normed Vector Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.
Let $X^\ast$ be the vector space of bounded linear functionals on $X$.
Let $A \subseteq X$ be a subset.
Let $\vee A$ be the closed linear span of $A$, i.e. the closure of the linear span of $A$.
Then $z \in \vee A$... | === Necessary condition ===
Let $z \in \vee A$.
Let $\ell \in X^\ast$ such that $\ell \restriction_A = 0$.
Recall, by {{Defof|Bounded Linear Functional|$X^\ast$}}, there is a $C > 0$ such that:
:$(1):\quad \forall x \in X : \size {\map \ell x} \le C \norm x$
On the other hand::
{{begin-eqn}}
{{eqn | l = \map \ell {z - ... | 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:Vector Space of Bounded Linear Functionals|vector space of bounded linear functionals]] on $X$.
Let $A \subseteq X$ be a [[Definition:Subset|... | === Necessary condition ===
Let $z \in \vee A$.
Let $\ell \in X^\ast$ such that $\ell \restriction_A = 0$.
Recall, by {{Defof|Bounded Linear Functional|$X^\ast$}}, there is a $C > 0$ such that:
:$(1):\quad \forall x \in X : \size {\map \ell x} \le C \norm x$
On the other hand::
{{begin-eqn}}
{{eqn | l = \map \ell {... | Spanning Criterion of Normed Vector Space | https://proofwiki.org/wiki/Spanning_Criterion_of_Normed_Vector_Space | https://proofwiki.org/wiki/Spanning_Criterion_of_Normed_Vector_Space | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Space of Bounded Linear Functionals/Vector Space",
"Definition:Subset",
"Definition:Closed Linear Span/Topological Vector Space",
"Definition:Closure (Topology)",
"Definition:Generated Submodule/Linear Span",
"Definition:Restriction/Mapping"
] | [] |
proofwiki-20970 | Absolute Error of Sum is not Greater than Sum of Absolute Errors | Let $X_1, X_2, \ldots, X_n$ be approximations to a collection of (true) values $x_1, x_2, \ldots, x_n$ respectively.
Let $\Delta X_i$ be the absolute error of $X_i$ in $x_i$ for $i \in \set {1, 2, \ldots, n}$.
Then:
:$\ds \map \Delta {\sum_{i \mathop = 1}^n X_i} \le \sum_{i \mathop = 1}^n \Delta X_i$
That is, the absol... | The proof proceeds by induction.
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the proposition:
:$\ds \map \Delta {\sum_{i \mathop = 1}^n X_i} \le \sum_{i \mathop = 1}^n \Delta X_i$
for appropriately defined $X_1, X_2, \ldots, X_n$ and $x_1, x_2, \ldots, x_n$. | Let $X_1, X_2, \ldots, X_n$ be [[Definition:Approximation|approximations]] to a [[Definition:Collection|collection]] of [[Definition:True Value|(true) values]] $x_1, x_2, \ldots, x_n$ respectively.
Let $\Delta X_i$ be the [[Definition:Absolute Error|absolute error]] of $X_i$ in $x_i$ for $i \in \set {1, 2, \ldots, n}$... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \map \Delta {\sum_{i \mathop = 1}^n X_i} \le \sum_{i \mathop = 1}^n \Delta X_i$
for appropriately defined $X_1, X_2, \ldots, X_n$ and $x_1, x_2, \ldo... | Absolute Error of Sum is not Greater than Sum of Absolute Errors | https://proofwiki.org/wiki/Absolute_Error_of_Sum_is_not_Greater_than_Sum_of_Absolute_Errors | https://proofwiki.org/wiki/Absolute_Error_of_Sum_is_not_Greater_than_Sum_of_Absolute_Errors | [
"Absolute Error"
] | [
"Definition:Approximation",
"Definition:Collection",
"Definition:True Value",
"Definition:Error/Absolute",
"Definition:Error/Absolute",
"Definition:Addition",
"Definition:Approximation",
"Definition:Greater Than",
"Definition:Addition",
"Definition:Error/Absolute",
"Definition:Approximation"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-20971 | Weak-* Dense Subset of Normed Dual Space Separates Points | Let $\GF \in \set {\R, \C}$.
Let $X$ be a normed vector space over $\GF$.
Let $X^\ast$ be the normed dual space of $X$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.
Let $D$ be everywhere dense in $\struct {X^\ast, w^\ast}$.
Then $D$ separates points. | Suppose that $D$ is everywhere dense in $\struct {X^\ast, w^\ast}$.
Let $x, y \in X$ be such that:
:$\map f x = \map g x$ for each $f, g \in D$.
Then:
:$\map {x^\wedge} f = \map {x^\wedge} g$ for each $f, g \in D$.
From Characterization of Continuity of Linear Functional in Weak-* Topology, $x^\wedge : \struct {X^\a... | Let $\GF \in \set {\R, \C}$.
Let $X$ 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 $w^\ast$ be the [[Definition:Weak-* Topology|weak-$\ast$ topology on $X^\ast$]].
Let $D$ be [[Definition:Everywhere Dense|... | Suppose that $D$ is [[Definition:Everywhere Dense|everywhere dense]] in $\struct {X^\ast, w^\ast}$.
Let $x, y \in X$ be such that:
:$\map f x = \map g x$ for each $f, g \in D$.
Then:
:$\map {x^\wedge} f = \map {x^\wedge} g$ for each $f, g \in D$.
From [[Characterization of Continuity of Linear Functional in Wea... | Weak-* Dense Subset of Normed Dual Space Separates Points | https://proofwiki.org/wiki/Weak-*_Dense_Subset_of_Normed_Dual_Space_Separates_Points | https://proofwiki.org/wiki/Weak-*_Dense_Subset_of_Normed_Dual_Space_Separates_Points | [
"Weak-* Topologies"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Definition:Everywhere Dense",
"Definition:Mappings Separating Points"
] | [
"Definition:Everywhere Dense",
"Characterization of Continuity of Linear Functional in Weak-* Topology",
"Definition:Continuous Mapping",
"Metric Space is T2",
"Definition:T2 Space",
"Definition:Everywhere Dense",
"Continuous Mappings into Hausdorff Space coinciding on Everywhere Dense Set coincide",
... |
proofwiki-20972 | Characterization of Paracompactness in T3 Space/Lemma 19 | :$\UU$ is a open cover of $X$ in $T$. | Let $s \in X$.
By definition of discrete:
:$\exists U \in \tau : x \in U : \size {\set{B \in \BB : U \cap B} } \le 1$
Hence:
:$U \in \UU$
Since $x$ was arbitrary:
:$\forall x \in X : \exists U \in \UU : x \in U$
It follows that $\UU$ is an open cover of $X$ in $T$ by definition.
{{qed}}
Category:Characterization of Par... | :$\UU$ is a [[Definition:Open Cover|open cover]] of $X$ in $T$. | Let $s \in X$.
By definition of [[DEfinition:Discrete Set of Subsets|discrete]]:
:$\exists U \in \tau : x \in U : \size {\set{B \in \BB : U \cap B} } \le 1$
Hence:
:$U \in \UU$
Since $x$ was arbitrary:
:$\forall x \in X : \exists U \in \UU : x \in U$
It follows that $\UU$ is an [[Definition:Open Cover|open cove... | Characterization of Paracompactness in T3 Space/Lemma 19 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_19 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_19 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Open Cover"
] | [
"DEfinition:Discrete Set of Subsets",
"Definition:Open Cover",
"Category:Characterization of Paracompactness in T3 Space"
] |
proofwiki-20973 | Vector Subspace of Normed Dual Space is Weak-* Dense iff Separates Points | Let $\GF \in \set {\R, \C}$.
Let $X$ be a normed vector space over $\GF$.
Let $X^\ast$ be the normed dual space of $X$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.
Let $F \subseteq X^\ast$ be a vector subspace of $X^\ast$.
Then $F$ is everywhere dense in $\struct {X^\ast, w^\ast}$ {{iff}} it separates point... | === Necessary Condition ===
Let $F$ be everywhere dense in $\struct {X^\ast, w^\ast}$.
From Weak-* Dense Subset of Normed Dual Space Separates Points, $F$ separates points.
{{qed|lemma}} | Let $\GF \in \set {\R, \C}$.
Let $X$ 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 $w^\ast$ be the [[Definition:Weak-* Topology|weak-$\ast$ topology on $X^\ast$]].
Let $F \subseteq X^\ast$ be a [[Definitio... | === Necessary Condition ===
Let $F$ be [[Definition:Everywhere Dense|everywhere dense]] in $\struct {X^\ast, w^\ast}$.
From [[Weak-* Dense Subset of Normed Dual Space Separates Points]], $F$ [[Definition:Mappings Separating Points|separates points]].
{{qed|lemma}} | Vector Subspace of Normed Dual Space is Weak-* Dense iff Separates Points | https://proofwiki.org/wiki/Vector_Subspace_of_Normed_Dual_Space_is_Weak-*_Dense_iff_Separates_Points | https://proofwiki.org/wiki/Vector_Subspace_of_Normed_Dual_Space_is_Weak-*_Dense_iff_Separates_Points | [
"Weak-* Topologies",
"Normed Dual Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Definition:Vector Subspace",
"Definition:Everywhere Dense",
"Definition:Mappings Separating Points"
] | [
"Definition:Everywhere Dense",
"Weak-* Dense Subset of Normed Dual Space Separates Points",
"Definition:Mappings Separating Points",
"Definition:Mappings Separating Points",
"Definition:Everywhere Dense",
"Definition:Mappings Separating Points",
"Definition:Everywhere Dense"
] |
proofwiki-20974 | Characterization of Paracompactness in T3 Space/Lemma 21 | :$\forall B \in \BB, x \in X : \map W x \cap W \sqbrk B \ne \O \leadsto \map {W \circ W} x \cap B \ne \O$ | Let $B \in \BB$.
Let $x \in X$.
Let $y \in \map W x \cap W \sqbrk B$.
By definition of intersection:
:$y \in \map W x$
and
:$y \in W \sqbrk B$
By definition of image of element:
:$\tuple{x, y} \in W$
By definition of symmetric:
:$\tuple{y, x} \in W$
By definition of image of subset:
:$\exists z \in B : \tuple{z, y} \in... | :$\forall B \in \BB, x \in X : \map W x \cap W \sqbrk B \ne \O \leadsto \map {W \circ W} x \cap B \ne \O$ | Let $B \in \BB$.
Let $x \in X$.
Let $y \in \map W x \cap W \sqbrk B$.
By definition of [[Definition:Set Intersection|intersection]]:
:$y \in \map W x$
and
:$y \in W \sqbrk B$
By definition of [[Definition:Image of Element under Relation|image of element]]:
:$\tuple{x, y} \in W$
By definition of [[Definition:Symm... | Characterization of Paracompactness in T3 Space/Lemma 21 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_21 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_21 | [
"Characterization of Paracompactness in T3 Space"
] | [] | [
"Definition:Set Intersection",
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Symmetric Relation",
"Definition:Image (Set Theory)/Relation/Subset",
"Definition:Composition of Relations",
"Definition:Image (Set Theory)/Relation/Element",
"Category:Characterization of Paracompactness in T3 ... |
proofwiki-20975 | Injective Linear Transformation between Normed Vector Spaces sends Closed Unit Ball to Closed Unit Ball iff Isometric Isomorphism | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ be a injective linear transformation.
Let $B_X^-$ be the closed unit ball of $X$.
Let $B_Y^-$ be the closed unit ball of $Y$.
Then $T$ is an isometric is... | === Necessary Condition ===
Suppose that $T$ is an isometric isomorphism.
Then:
:$\norm {T x}_Y = \norm x_X$ for each $x \in X$.
Hence if $\norm x_X \le 1$, we have $\norm {T x}_Y \le 1$.
So $T \sqbrk {B_X^-} \subseteq B_Y^-$.
Conversely, let $y \in B_Y^-$.
Since $T$ is an isometric isomorphism:
:$T$ is bijective and... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ be a [[Definition:Injection|injective]] [[Definition:Linear Transformation|linear transformation]].
Let $B_X^-$ be t... | === Necessary Condition ===
Suppose that $T$ is an [[Definition:Isometric Isomorphism on Normed Vector Space|isometric isomorphism]].
Then:
:$\norm {T x}_Y = \norm x_X$ for each $x \in X$.
Hence if $\norm x_X \le 1$, we have $\norm {T x}_Y \le 1$.
So $T \sqbrk {B_X^-} \subseteq B_Y^-$.
Conversely, let $y \in B_Y... | Injective Linear Transformation between Normed Vector Spaces sends Closed Unit Ball to Closed Unit Ball iff Isometric Isomorphism | https://proofwiki.org/wiki/Injective_Linear_Transformation_between_Normed_Vector_Spaces_sends_Closed_Unit_Ball_to_Closed_Unit_Ball_iff_Isometric_Isomorphism | https://proofwiki.org/wiki/Injective_Linear_Transformation_between_Normed_Vector_Spaces_sends_Closed_Unit_Ball_to_Closed_Unit_Ball_iff_Isometric_Isomorphism | [
"Isometric Isomorphisms (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Injection",
"Definition:Linear Transformation",
"Definition:Closed Unit Ball",
"Definition:Closed Unit Ball",
"Definition:Isometric Isomorphism/Normed Vector Space"
] | [
"Definition:Isometric Isomorphism/Normed Vector Space",
"Definition:Isometric Isomorphism/Normed Vector Space",
"Definition:Bijection",
"Definition:Linear Isometry",
"Definition:Linear Isometry",
"Definition:Isometric Isomorphism/Normed Vector Space"
] |
proofwiki-20976 | Image of Point under Open Neighborhood of Diagonal is Open Neighborhood of Point | Let $T = \struct{X, \tau}$ be a topological space.
Let $\tau_{X \times X}$ denote the product topology on the cartesian product $X \times X$.
Let $U$ be an open neighborhood of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct {X \times X, \tau_{X \times X} }$.
Then:
:$\forall x \in X : \map U x$ is... | Let $x \in X$.
By definition of open neighborhood:
:$U$ is a neighborhood of $\Delta_X$
From Image of Point under Neighborhood of Diagonal is Neighborhood of Point:
:$\map U x$ is a neighborhood of $x$ in $T$
It remains to show that $\map U x$ is open in $T$, that is, $\map U x \in \tau$.
Let:
:$\WW = \set{W \in \tau :... | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\tau_{X \times X}$ denote the [[Definition:Product Topology|product topology]] on the [[Definition:Cartesian Product|cartesian product]] $X \times X$.
Let $U$ be an [[Definition:Open Neighborhood|open neighborhood]] of the [[Def... | Let $x \in X$.
By definition of [[Definition:Open Neighborhood|open neighborhood]]:
:$U$ is a [[Definition:Neighborhood of Set|neighborhood]] of $\Delta_X$
From [[Image of Point under Neighborhood of Diagonal is Neighborhood of Point]]:
:$\map U x$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$ in $T$... | Image of Point under Open Neighborhood of Diagonal is Open Neighborhood of Point | https://proofwiki.org/wiki/Image_of_Point_under_Open_Neighborhood_of_Diagonal_is_Open_Neighborhood_of_Point | https://proofwiki.org/wiki/Image_of_Point_under_Open_Neighborhood_of_Diagonal_is_Open_Neighborhood_of_Point | [
"Open Sets",
"Product Spaces",
"Relations"
] | [
"Definition:Topological Space",
"Definition:Product Topology",
"Definition:Cartesian Product",
"Definition:Open Neighborhood",
"Definition:Diagonal Relation",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Open Neighborhood"
] | [
"Definition:Open Neighborhood",
"Definition:Neighborhood (Topology)/Set",
"Image of Point under Neighborhood of Diagonal is Neighborhood of Point",
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Set/Topology",
"Definition:Image (Set Theory)/Relation/Element",
"Definition:Product Topology",... |
proofwiki-20977 | Image of Subset under Open Neighborhood of Diagonal is Open Neighborhood of Subset | Let $T = \struct{X, \tau}$ be a topological space.
Let $\tau_{X \times X}$ denote the product topology on the cartesian product $X \times X$.
Let $U$ be an open neighborhood of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct {X \times X, \tau_{X \times X} }$.
Then:
:$\forall A \subseteq X : U \sqb... | Let $A \subseteq X$.
From Image of Subset under Neighborhood of Diagonal is Neighborhood of Subset:
:$U \sqbrk A$ is a neighborhood of $A$ in $T$
From Image of Subset under Relation equals Union of Images of Elements:
:$U \sqbrk A = \ds \bigcup_{x \in A} \map U x$
From Image of Point under Open Neighborhood of Diagonal... | Let $T = \struct{X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\tau_{X \times X}$ denote the [[Definition:Product Topology|product topology]] on the [[Definition:Cartesian Product|cartesian product]] $X \times X$.
Let $U$ be an [[Definition:Open Neighborhood|open neighborhood]] of the [[Def... | Let $A \subseteq X$.
From [[Image of Subset under Neighborhood of Diagonal is Neighborhood of Subset]]:
:$U \sqbrk A$ is a [[Definition:Neighborhood of Set|neighborhood]] of $A$ in $T$
From [[Image of Subset under Relation equals Union of Images of Elements]]:
:$U \sqbrk A = \ds \bigcup_{x \in A} \map U x$
From [... | Image of Subset under Open Neighborhood of Diagonal is Open Neighborhood of Subset | https://proofwiki.org/wiki/Image_of_Subset_under_Open_Neighborhood_of_Diagonal_is_Open_Neighborhood_of_Subset | https://proofwiki.org/wiki/Image_of_Subset_under_Open_Neighborhood_of_Diagonal_is_Open_Neighborhood_of_Subset | [
"Open Sets",
"Product Spaces",
"Relations"
] | [
"Definition:Topological Space",
"Definition:Product Topology",
"Definition:Cartesian Product",
"Definition:Open Neighborhood",
"Definition:Diagonal Relation",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Open Neighborhood"
] | [
"Image of Subset under Neighborhood of Diagonal is Neighborhood of Subset",
"Definition:Neighborhood (Topology)/Set",
"Image of Subset under Relation equals Union of Images of Elements",
"Image of Point under Open Neighborhood of Diagonal is Open Neighborhood of Point",
"Definition:Open Neighborhood/Point",... |
proofwiki-20978 | Primitive of Reciprocal of a x squared + b by Root of c x squared + d/Lemma | :$\ds \int \dfrac {\d x} {\paren {a x^2 + b} \sqrt {c x^2 + d} } = \dfrac {\sqrt c} {2 a} \int \dfrac {\d u} {\paren {\sqrt {\paren {u - \frac d 2}^2 - \frac d 4} } \paren {u - \frac {a d + b c} a} }$
where $u := c x^2 + d$. | {{begin-eqn}}
{{eqn | l = u
| r = c x^2 + d
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \sqrt {\dfrac {u - d} c}
| c =
}}
{{eqn | lo= \text {and}
| l = \frac {\d u} {\d x}
| r = 2 c x
| c = Primitive of Power
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \dfrac {... | :$\ds \int \dfrac {\d x} {\paren {a x^2 + b} \sqrt {c x^2 + d} } = \dfrac {\sqrt c} {2 a} \int \dfrac {\d u} {\paren {\sqrt {\paren {u - \frac d 2}^2 - \frac d 4} } \paren {u - \frac {a d + b c} a} }$
where $u := c x^2 + d$. | {{begin-eqn}}
{{eqn | l = u
| r = c x^2 + d
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = \sqrt {\dfrac {u - d} c}
| c =
}}
{{eqn | lo= \text {and}
| l = \frac {\d u} {\d x}
| r = 2 c x
| c = [[Primitive of Power]]
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \int \d... | Primitive of Reciprocal of a x squared + b by Root of c x squared + d/Lemma | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_+_b_by_Root_of_c_x_squared_+_d/Lemma | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_x_squared_+_b_by_Root_of_c_x_squared_+_d/Lemma | [
"Primitive of Reciprocal of a x squared + b by Root of c x squared + d"
] | [] | [
"Primitive of Power",
"Integration by Substitution",
"Category:Primitive of Reciprocal of a x squared + b by Root of c x squared + d"
] |
proofwiki-20979 | Argument of Complex Conjugate equals Negative of Argument | Let $z \in \C$ be a complex number.
Then:
:$\arg {\overline z} = -\arg z$
where:
:$\arg$ denotes the argument of a complex number
:$\overline z$ denotes the complex conjugate of $z$. | Let $z$ be expressed in polar form:
:$z := r \paren {\cos \theta + i \sin \theta}$
Then:
{{begin-eqn}}
{{eqn | l = \overline z
| r = r \paren {\cos \theta - i \sin \theta}
| c = Polar Form of Complex Conjugate
}}
{{eqn | r = r \paren {\map \cos {-\theta} + i \map \sin {-\theta} }
| c = Cosine Function... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\arg {\overline z} = -\arg z$
where:
:$\arg$ denotes the [[Definition:Argument of Complex Number|argument]] of a [[Definition:Complex Number|complex number]]
:$\overline z$ denotes the [[Definition:Complex Conjugate|complex conjugate]] of $z$. | Let $z$ be expressed in [[Definition:Polar Form of Complex Number|polar form]]:
:$z := r \paren {\cos \theta + i \sin \theta}$
Then:
{{begin-eqn}}
{{eqn | l = \overline z
| r = r \paren {\cos \theta - i \sin \theta}
| c = [[Polar Form of Complex Conjugate]]
}}
{{eqn | r = r \paren {\map \cos {-\theta} +... | Argument of Complex Conjugate equals Negative of Argument | https://proofwiki.org/wiki/Argument_of_Complex_Conjugate_equals_Negative_of_Argument | https://proofwiki.org/wiki/Argument_of_Complex_Conjugate_equals_Negative_of_Argument | [
"Complex Conjugates"
] | [
"Definition:Complex Number",
"Definition:Argument of Complex Number",
"Definition:Complex Number",
"Definition:Complex Conjugate"
] | [
"Definition:Complex Number/Polar Form",
"Polar Form of Complex Conjugate",
"Cosine Function is Even",
"Sine Function is Odd",
"Definition:Argument of Complex Number",
"Definition:Complex Number"
] |
proofwiki-20980 | Complex Division as Product with Conjugate over Square of Modulus | Let $z_1$ and $z_2$ be complex numbers.
Then the operation of division can be expressed as:
:$\dfrac {z_1} {z_2} = \dfrac {z_1 \overline {z_2} } {\cmod {z_2}^2}$
where:
:$\overline {z_2}$ denotes the complex conjugate of $z_2$
:$\cmod {z_2}$ denotes the complex modulus of $z_2$. | {{begin-eqn}}
{{eqn | l = \dfrac {z_1} {z_2}
| r = \dfrac {z_1 \overline {z_2} } {z_2 \overline {z_2} }
| c = multiplying top and bottom by $\overline {z_2}$
}}
{{eqn | r = \dfrac {z_1 \overline {z_2} } {\cmod {z_2}^2}
| c = Modulus in Terms of Conjugate
}}
{{end-eqn}}
{{qed}} | Let $z_1$ and $z_2$ be [[Definition:Complex Number|complex numbers]].
Then the [[Definition:Binary Operation|operation]] of [[Definition:Complex Division|division]] can be expressed as:
:$\dfrac {z_1} {z_2} = \dfrac {z_1 \overline {z_2} } {\cmod {z_2}^2}$
where:
:$\overline {z_2}$ denotes the [[Definition:Complex Conj... | {{begin-eqn}}
{{eqn | l = \dfrac {z_1} {z_2}
| r = \dfrac {z_1 \overline {z_2} } {z_2 \overline {z_2} }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $\overline {z_2}$
}}
{{eqn | r = \dfrac {z_1 \overline {z_2} } {\cmod {z_2}^2}
| c = [[Modulus in Terms of Con... | Complex Division as Product with Conjugate over Square of Modulus | https://proofwiki.org/wiki/Complex_Division_as_Product_with_Conjugate_over_Square_of_Modulus | https://proofwiki.org/wiki/Complex_Division_as_Product_with_Conjugate_over_Square_of_Modulus | [
"Complex Division",
"Complex Conjugates",
"Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Operation/Binary Operation",
"Definition:Division/Field/Complex Numbers",
"Definition:Complex Conjugate",
"Definition:Complex Modulus"
] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Modulus in Terms of Conjugate"
] |
proofwiki-20981 | Square of Complex Number | Let $z = a + i b$ be a complex number.
Then its square is given by:
:$z^2 = a^2 - b^2 + i \paren {2 a b}$ | {{begin-eqn}}
{{eqn | l = z^2
| r = \paren {a + i b}^2
| c = {{hypothesis}}
}}
{{eqn | r = \paren {a + i b} \paren {a + i b}
| c = {{Defof|Square Function}}
}}
{{eqn | r = \paren {a \times a - b \times b} + i \paren {a \times b + b \times a}
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = ... | Let $z = a + i b$ be a [[Definition:Complex Number|complex number]].
Then its [[Definition:Square Function|square]] is given by:
:$z^2 = a^2 - b^2 + i \paren {2 a b}$ | {{begin-eqn}}
{{eqn | l = z^2
| r = \paren {a + i b}^2
| c = {{hypothesis}}
}}
{{eqn | r = \paren {a + i b} \paren {a + i b}
| c = {{Defof|Square Function}}
}}
{{eqn | r = \paren {a \times a - b \times b} + i \paren {a \times b + b \times a}
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = ... | Square of Complex Number | https://proofwiki.org/wiki/Square_of_Complex_Number | https://proofwiki.org/wiki/Square_of_Complex_Number | [
"Complex Numbers",
"Square Function"
] | [
"Definition:Complex Number",
"Definition:Square/Function"
] | [] |
proofwiki-20982 | Cube of Complex Number | Let $z = a + i b$ be a complex number.
Then its cube is given by:
:$z^3 = a^3 - 3 a b^2 + i \paren {3 a^2 b - b^3}$ | {{begin-eqn}}
{{eqn | l = z^3
| r = \paren {a + i b}^3
| c = {{hypothesis}}
}}
{{eqn | r = \paren {a + i b}^2 \paren {a + i b}
| c = {{Defof|Cube (Algebra)}}
}}
{{eqn | r = \paren {a^2 - b^2 + i \paren {2 a b} } \paren {a + i b}
| c = Square of Complex Number
}}
{{eqn | r = \paren {\paren {a^2 -... | Let $z = a + i b$ be a [[Definition:Complex Number|complex number]].
Then its [[Definition:Cube (Algebra)|cube]] is given by:
:$z^3 = a^3 - 3 a b^2 + i \paren {3 a^2 b - b^3}$ | {{begin-eqn}}
{{eqn | l = z^3
| r = \paren {a + i b}^3
| c = {{hypothesis}}
}}
{{eqn | r = \paren {a + i b}^2 \paren {a + i b}
| c = {{Defof|Cube (Algebra)}}
}}
{{eqn | r = \paren {a^2 - b^2 + i \paren {2 a b} } \paren {a + i b}
| c = [[Square of Complex Number]]
}}
{{eqn | r = \paren {\paren {a... | Cube of Complex Number | https://proofwiki.org/wiki/Cube_of_Complex_Number | https://proofwiki.org/wiki/Cube_of_Complex_Number | [
"Complex Numbers",
"Cube Function"
] | [
"Definition:Complex Number",
"Definition:Cube/Algebra"
] | [
"Square of Complex Number"
] |
proofwiki-20983 | Fourth Power of Complex Number | Let $z = a + i b$ be a complex number.
Then its fourth power is given by:
:$z^4 = a^4 - 6 a^2 b^2 + b^4 + i \paren {4 a^3 b - 4 a b^3}$ | {{begin-eqn}}
{{eqn | l = z^4
| r = \paren {a + i b}^4
| c = {{hypothesis}}
}}
{{eqn | r = a^4 + 4 a^3 \paren {i b} + 6 a^2 \paren {i b}^2 + 4 a \paren {i b}^3 + \paren {i b}^4
| c = Fourth Power of Sum
}}
{{eqn | r = a^4 + 4 i a^3 b + 6 i^2 a^2 b^2 + 4 i^3 a b^3 + i^4 b^4
| c = Complex Multipli... | Let $z = a + i b$ be a [[Definition:Complex Number|complex number]].
Then its [[Definition:Fourth Power|fourth power]] is given by:
:$z^4 = a^4 - 6 a^2 b^2 + b^4 + i \paren {4 a^3 b - 4 a b^3}$ | {{begin-eqn}}
{{eqn | l = z^4
| r = \paren {a + i b}^4
| c = {{hypothesis}}
}}
{{eqn | r = a^4 + 4 a^3 \paren {i b} + 6 a^2 \paren {i b}^2 + 4 a \paren {i b}^3 + \paren {i b}^4
| c = [[Fourth Power of Sum]]
}}
{{eqn | r = a^4 + 4 i a^3 b + 6 i^2 a^2 b^2 + 4 i^3 a b^3 + i^4 b^4
| c = [[Complex Mu... | Fourth Power of Complex Number | https://proofwiki.org/wiki/Fourth_Power_of_Complex_Number | https://proofwiki.org/wiki/Fourth_Power_of_Complex_Number | [
"Complex Numbers",
"Fourth Powers"
] | [
"Definition:Complex Number",
"Definition:Fourth Power"
] | [
"Binomial Theorem/Examples/4th Power of Sum",
"Complex Multiplication is Commutative"
] |
proofwiki-20984 | Fifth Power of Complex Number | Let $z = a + i b$ be a complex number.
Then its fifth power is given by:
:$z^5 = a^5 - 10 a^3 b^2 + 5 a b^4 + i \paren {5 a^4 b - 10 a^2 b^3 + b^5}$ | {{begin-eqn}}
{{eqn | l = z^5
| r = \paren {a + i b}^5
| c = {{hypothesis}}
}}
{{eqn | r = a^5 + 5 a^4 \paren {i b} + 10 a^3 \paren {i b}^2 + 10 a^2 \paren {i b}^3 + 5 a \paren {i b}^4 + \paren {i b}^5
| c = Fifth Power of Sum
}}
{{eqn | r = a^5 + 5 i a^4 b + 10 i^2 a^3 b^2 + 10 i^3 a^2 b^3 + 5 i^4 a ... | Let $z = a + i b$ be a [[Definition:Complex Number|complex number]].
Then its [[Definition:Fifth Power|fifth power]] is given by:
:$z^5 = a^5 - 10 a^3 b^2 + 5 a b^4 + i \paren {5 a^4 b - 10 a^2 b^3 + b^5}$ | {{begin-eqn}}
{{eqn | l = z^5
| r = \paren {a + i b}^5
| c = {{hypothesis}}
}}
{{eqn | r = a^5 + 5 a^4 \paren {i b} + 10 a^3 \paren {i b}^2 + 10 a^2 \paren {i b}^3 + 5 a \paren {i b}^4 + \paren {i b}^5
| c = [[Fifth Power of Sum]]
}}
{{eqn | r = a^5 + 5 i a^4 b + 10 i^2 a^3 b^2 + 10 i^3 a^2 b^3 + 5 i^... | Fifth Power of Complex Number | https://proofwiki.org/wiki/Fifth_Power_of_Complex_Number | https://proofwiki.org/wiki/Fifth_Power_of_Complex_Number | [
"Complex Numbers",
"Fifth Powers"
] | [
"Definition:Complex Number",
"Definition:Fifth Power"
] | [
"Binomial Theorem/Examples/5th Power of Sum",
"Complex Multiplication is Commutative"
] |
proofwiki-20985 | Riesz-Kakutani Representation Theorem | Let $X$ be a Hausdorff compact space.
Let $\map \BB X$ be the Borel $\sigma$-algebra on $X$.
Let $\map C {X, \R}$ be the space of real-valued continuous functions.
Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $X$.
Let $\struct {C', \norm {\, \cdot \,}_{C'} }$ be the normed dual space of $\struct {\map C {X,... | {{ProofWanted}}
{{Namedfor|Frigyes Riesz|cat = Riesz F|name2 = Shizuo Kakutani|cat2 = Kakutani}} | Let $X$ be a [[Definition:Hausdorff Space|Hausdorff]] [[Definition:Compact Topological Space|compact space]].
Let $\map \BB X$ be the [[Definition:Borel Sigma-Algebra/Topological Space|Borel $\sigma$-algebra]] on $X$.
Let $\map C {X, \R}$ be the [[Definition:Vector Space|space]] of [[Definition:Real-Valued Function|r... | {{ProofWanted}}
{{Namedfor|Frigyes Riesz|cat = Riesz F|name2 = Shizuo Kakutani|cat2 = Kakutani}} | Riesz-Kakutani Representation Theorem | https://proofwiki.org/wiki/Riesz-Kakutani_Representation_Theorem | https://proofwiki.org/wiki/Riesz-Kakutani_Representation_Theorem | [
"Functional Analysis",
"Riesz-Markov-Kakutani Representation Theorem"
] | [
"Definition:T2 Space",
"Definition:Compact Topological Space",
"Definition:Borel Sigma-Algebra/Topological Space",
"Definition:Vector Space",
"Definition:Real-Valued Function",
"Definition:Continuous Mapping",
"Definition:Supremum Norm",
"Definition:Normed Dual Space",
"Definition:Signed Measure",
... | [] |
proofwiki-20986 | Arbitrary Power of Complex Number | Let $z = a + i b$ be a complex number.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
{{begin-eqn}}
{{eqn | l = z^n
| r = \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom n j a^{n - j} b^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop ... | === Lemma ===
{{:Arbitrary Power of Complex Number/Lemma}}{{qed|lemma}}
The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\ds z^n = \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom n j a^{n - j} b^j} + i \paren {\sum_... | Let $z = a + i b$ be a [[Definition:Complex Number|complex number]].
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = z^n
| r = \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbino... | === [[Arbitrary Power of Complex Number/Lemma|Lemma]] ===
{{:Arbitrary Power of Complex Number/Lemma}}{{qed|lemma}}
The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds z^n = \paren {\sum_{\substack {0 \m... | Arbitrary Power of Complex Number | https://proofwiki.org/wiki/Arbitrary_Power_of_Complex_Number | https://proofwiki.org/wiki/Arbitrary_Power_of_Complex_Number | [
"Arbitrary Power of Complex Number",
"Complex Numbers",
"Powers"
] | [
"Definition:Complex Number",
"Definition:Strictly Positive/Integer"
] | [
"Arbitrary Power of Complex Number/Lemma",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Arbitrary Power of Complex Number",
"Arbitrary Power of Complex Number",
"Principle of Mathematical Induction"
] |
proofwiki-20987 | Neumann Series Theorem | Let $X$ be a Banach space.
Let $\map {CL} X$ be the continous linear transformation space.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Let $A \in \map {CL} X$ be such that $\norm A < 1$.
Let $\circ$ be the composition of mappings.
Let $I$ be the identity mapping.
Then:
:$I - A$ is invertible in $\map {CL} ... | Let $\ds S_k := \sum_{n \mathop = 0}^k A^n$. | Let $X$ be a [[Definition:Banach Space|Banach space]].
Let $\map {CL} X$ be the [[Definition:Continuous Linear Transformation Space|continous linear transformation space]].
Let $\norm {\, \cdot \,}$ be the [[Definition:Supremum Operator Norm|supremum operator norm]].
Let $A \in \map {CL} X$ be such that $\norm A < 1... | Let $\ds S_k := \sum_{n \mathop = 0}^k A^n$. | Neumann Series Theorem | https://proofwiki.org/wiki/Neumann_Series_Theorem | https://proofwiki.org/wiki/Neumann_Series_Theorem | [
"Neumann Series",
"Continuous Linear Transformations",
"Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Supremum Operator Norm",
"Definition:Composition of Mappings",
"Definition:Identity Mapping",
"Definition:Invertible Continuous Linear Operator"
] | [] |
proofwiki-20988 | Image of Bounded Linear Transformation is Everywhere Dense iff Dual Operator is Injective | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be normed vector spaces over $\GF$.
Let $X^\ast$ and $Y^\ast$ be the normed dual spaces of $X$ and $Y$ respectively.
Let $T : X \to Y$ be a bounded linear transformation.
Let $T^\ast : Y^\ast \to X^\ast$ be the dual operator of $T$.
Then $T \sqbrk X$ is everywhere dense ... | From Annihilator of Image of Bounded Linear Transformation is Kernel of Dual Operator, we have:
:$T \sqbrk X^\bot = \map \ker {T^\ast}$
where $T \sqbrk X^\bot$ denotes the annihilator of $T \sqbrk X$.
From Linear Transformation is Injective iff Kernel Contains Only Zero, we then have that $T^\ast$ is injective {{iff}}:... | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $X^\ast$ and $Y^\ast$ be the [[Definition:Normed Dual Space|normed dual spaces]] of $X$ and $Y$ respectively.
Let $T : X \to Y$ be a [[Definition:Bounded Linear Transformation|bounded linear tra... | From [[Annihilator of Image of Bounded Linear Transformation is Kernel of Dual Operator]], we have:
:$T \sqbrk X^\bot = \map \ker {T^\ast}$
where $T \sqbrk X^\bot$ denotes the [[Definition:Annihilator of Subspace of Banach Space|annihilator]] of $T \sqbrk X$.
From [[Linear Transformation is Injective iff Kernel Contai... | Image of Bounded Linear Transformation is Everywhere Dense iff Dual Operator is Injective/Proof 2 | https://proofwiki.org/wiki/Image_of_Bounded_Linear_Transformation_is_Everywhere_Dense_iff_Dual_Operator_is_Injective | https://proofwiki.org/wiki/Image_of_Bounded_Linear_Transformation_is_Everywhere_Dense_iff_Dual_Operator_is_Injective/Proof_2 | [
"Dual Operators",
"Image of Bounded Linear Transformation is Everywhere Dense iff Dual Operator is Injective"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Bounded Linear Transformation",
"Definition:Dual Operator",
"Definition:Everywhere Dense",
"Definition:Injection"
] | [
"Annihilator of Image of Bounded Linear Transformation is Kernel of Dual Operator",
"Definition:Annihilator of Subspace of Banach Space",
"Linear Transformation is Injective iff Kernel Contains Only Zero",
"Definition:Injection",
"Annihilator of Subspace of Banach Space is Zero iff Subspace is Everywhere De... |
proofwiki-20989 | Offset URM Program is Primitive Recursive | There exists a primitive recursive function $\operatorname {Offset} : \N^2 \to \N$ such that, for $e, k \in \N$:
{{begin-itemize}}
{{item|*|If $e$ does not code a URM program, then $\map {\operatorname {Offset} } {e, k} {{=}} 0$.}}
{{item|*|If $e$ codes a URM program $P$, then $\map {\operatorname {Offset} } {e, k}$ co... | The basic instructions of $P'$ will be defined as:
:<nowiki>$P'_i = \begin {cases}
\map C {1, 1} & : i \le k \\ \\
P_{i - k} & : i > k \land P_{i - k} \notin \texttt {Jump} \\ \\
\map J {m, n, q + k} & : i > k \land P_{i - k} = \map J {m, n, q}
\end {cases}$</nowiki>
Property $(1)$ holds by the first case.
Property $(2... | There exists a [[Definition:Primitive Recursive Function|primitive recursive function]] $\operatorname {Offset} : \N^2 \to \N$ such that, for $e, k \in \N$:
{{begin-itemize}}
{{item|*|If $e$ does not [[Unique Code for URM Program|code a URM program]], then $\map {\operatorname {Offset} } {e, k} {{=}} 0$.}}
{{item|*|If... | The basic instructions of $P'$ will be defined as:
:<nowiki>$P'_i = \begin {cases}
\map C {1, 1} & : i \le k \\ \\
P_{i - k} & : i > k \land P_{i - k} \notin \texttt {Jump} \\ \\
\map J {m, n, q + k} & : i > k \land P_{i - k} = \map J {m, n, q}
\end {cases}$</nowiki>
Property $(1)$ holds by the first case.
Property $... | Offset URM Program is Primitive Recursive | https://proofwiki.org/wiki/Offset_URM_Program_is_Primitive_Recursive | https://proofwiki.org/wiki/Offset_URM_Program_is_Primitive_Recursive | [
"Primitive Recursive Functions",
"URM Programs"
] | [
"Definition:Primitive Recursive/Function",
"Unique Code for URM Program",
"Unique Code for URM Program",
"Definition:Unlimited Register Machine/Program/Basic Instruction",
"Definition:Unlimited Register Machine/Program/Line",
"Definition:Unlimited Register Machine/Program/Basic Instruction",
"Definition... | [
"Definition:Mathematical Induction",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Principle of Mathematical Induction/Zero-Based",
"Unique Code for URM Instruction",
"Definition:Primitive Recursive/Function",
"Definition by Cases is Primitive Recursive",
"Ordering Relation... |
proofwiki-20990 | Parameterization Theorem | Let $m, n \in \N$ be natural numbers.
There exists a primitive recursive function $S^m_n : \N^{1 + m} \to \N$ such that, for $e, x_1, \dotsc, x_m$:
* If $e$ is not the code for a URM program, $\map {S^m_n} {e, x_1, \dotsc, x_m} = 0$
* Otherwise, let $P$ be that program.
:Let $f$ be the $m + n$-arity function computed b... | Let $N$ be an abbreviation for:
:$m + n + x_1 + \dotso + x_m$
in the following definition.
Define:
:$\map {S^m_n} {e, x_1, \dotsc, x_m} = \begin{cases} {\map p 1}^{\map \beta {\map C {n, m + n}}} \times \dotso \times {\map p n}^{\map \beta {\map C {1, 1 + m}}} \times {\map p {1 + n}}^{\map \beta {\map Z 1}} \times \dot... | Let $m, n \in \N$ be [[Definition:Natural Number|natural numbers]].
There exists a [[Definition:Primitive Recursive Function|primitive recursive function]] $S^m_n : \N^{1 + m} \to \N$ such that, for $e, x_1, \dotsc, x_m$:
* If $e$ is not the [[Unique Code for URM Program|code]] for a [[Definition:URM Program|URM prog... | Let $N$ be an abbreviation for:
:$m + n + x_1 + \dotso + x_m$
in the following definition.
Define:
:$\map {S^m_n} {e, x_1, \dotsc, x_m} = \begin{cases} {\map p 1}^{\map \beta {\map C {n, m + n}}} \times \dotso \times {\map p n}^{\map \beta {\map C {1, 1 + m}}} \times {\map p {1 + n}}^{\map \beta {\map Z 1}} \times \do... | Parameterization Theorem | https://proofwiki.org/wiki/Parameterization_Theorem | https://proofwiki.org/wiki/Parameterization_Theorem | [
"Named Theorems",
"Primitive Recursive Functions",
"URM Programs"
] | [
"Definition:Natural Numbers",
"Definition:Primitive Recursive/Function",
"Unique Code for URM Program",
"Definition:Unlimited Register Machine/Program",
"Definition:URM Computability",
"Unique Code for URM Program",
"Definition:Unlimited Register Machine/Program",
"Definition:URM Computability"
] | [
"Unique Code for URM Instruction",
"Definition by Cases is Primitive Recursive",
"Set of Codes for URM Programs is Primitive Recursive",
"Constant Function is Primitive Recursive",
"Addition is Primitive Recursive",
"Multiplication is Primitive Recursive",
"Bounded Product is Primitive Recursive",
"Pr... |
proofwiki-20991 | Characterization of Paracompactness in T3 Space/Lemma 9 | Let $\BB$ be a discrete set of subsets of $X$.
Then there exists an open neighborhood $W$ of the diagonal $\Delta_X$ of $X \times X$ in $T \times T$:
:$\forall x \in X : \card {\set{B \in \BB : \map W x \cap W \sqbrk B \ne \O}} \le 1$ | Let:
:$\UU = \set{ U \in \tau : \card {\set{B \in \BB : U \cap B} } \le 1}$ | Let $\BB$ be a [[Definition:Discrete Set of Subsets|discrete set of subsets]] of $X$.
Then there exists an [[Definition:Open Neighborhood|open neighborhood]] $W$ of the [[Definition:Diagonal Relation|diagonal $\Delta_X$]] of $X \times X$ in $T \times T$:
:$\forall x \in X : \card {\set{B \in \BB : \map W x \cap W \sq... | Let:
:$\UU = \set{ U \in \tau : \card {\set{B \in \BB : U \cap B} } \le 1}$ | Characterization of Paracompactness in T3 Space/Lemma 9 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_9 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_9 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Discrete Set of Subsets",
"Definition:Open Neighborhood",
"Definition:Diagonal Relation"
] | [] |
proofwiki-20992 | Characterization of Paracompactness in T3 Space/Lemma 10 | :$\WW$ is an open $\sigma$-discrete refinement of $\UU$ | === $\WW$ is Set of Open Sets ===
Let:
:$W \in \WW$
By definition of $\WW$:
:$\exists n \in \N, A \in \AA : W = U_A \cap V_n \sqbrk A$
We have {{hypothesis}}:
:$U_A \in \tau$
From Image of Subset under Open Neighborhood of Diagonal is Open Neighborhood of Subset:
:$V_n \sqbrk A \in \tau$
By {{Open-set-axiom|2}}:
:$W \i... | :$\WW$ is an [[Definition:Open Sigma-Discrete Set of Subsets|open $\sigma$-discrete]] [[Definition:Refinement of Cover|refinement]] of $\UU$ | === $\WW$ is Set of Open Sets ===
Let:
:$W \in \WW$
By definition of $\WW$:
:$\exists n \in \N, A \in \AA : W = U_A \cap V_n \sqbrk A$
We have {{hypothesis}}:
:$U_A \in \tau$
From [[Image of Subset under Open Neighborhood of Diagonal is Open Neighborhood of Subset]]:
:$V_n \sqbrk A \in \tau$
By {{Open-set-axio... | Characterization of Paracompactness in T3 Space/Lemma 10 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_10 | https://proofwiki.org/wiki/Characterization_of_Paracompactness_in_T3_Space/Lemma_10 | [
"Characterization of Paracompactness in T3 Space"
] | [
"Definition:Open Sigma-Discrete Set of Subsets",
"Definition:Refinement of Cover"
] | [
"Image of Subset under Open Neighborhood of Diagonal is Open Neighborhood of Subset",
"Definition:Set of Sets",
"Definition:Open Set/Topology"
] |
proofwiki-20993 | Real and Imaginary Parts of Integer Power of Complex Number are Harmonic | Let $z \in \C$ be a complex number.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $z^n$ denote $z$ raised to the $n$th power.
Then both the real part $\map \Re {z^n}$ and the imaginary part $\map \Im {z^n}$ of $z^n$ are harmonic polynomials. | === Real Part of Integer Power of Complex Number is Harmonic ===
{{:Real Part of Integer Power of Complex Number is Harmonic}}{{qed|lemma}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $z^n$ denote $z$ raised to the [[Definition:Integer Power|$n$th power]].
Then both the [[Definition:Real Part|real part]] $\map \Re {z^n}$ and the [[De... | === [[Real Part of Integer Power of Complex Number is Harmonic]] ===
{{:Real Part of Integer Power of Complex Number is Harmonic}}{{qed|lemma}} | Real and Imaginary Parts of Integer Power of Complex Number are Harmonic | https://proofwiki.org/wiki/Real_and_Imaginary_Parts_of_Integer_Power_of_Complex_Number_are_Harmonic | https://proofwiki.org/wiki/Real_and_Imaginary_Parts_of_Integer_Power_of_Complex_Number_are_Harmonic | [
"Harmonic Polynomials",
"Complex Powers"
] | [
"Definition:Complex Number",
"Definition:Strictly Positive/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part",
"Definition:Harmonic Polynomial"
] | [
"Real Part of Integer Power of Complex Number is Harmonic"
] |
proofwiki-20994 | Characterization of Open Linear Transformation between Normed Vector Spaces | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ be a linear transformation.
Let $B_X^-$ and $B_Y^-$ be the closed unit balls of $X$ and $Y$ respectively.
{{TFAE}}
:$(1) \quad$ $T$ is an open map
:$(2)... | Let $B_X^O$ and $B_Y^O$ be the open unit balls of $X$ and $Y$ respectively. | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Let $B_X^-$ and $B_Y^-$ be the [[Definition:Closed... | Let $B_X^O$ and $B_Y^O$ be the [[Definition:Open Unit Ball|open unit balls]] of $X$ and $Y$ respectively. | Characterization of Open Linear Transformation between Normed Vector Spaces | https://proofwiki.org/wiki/Characterization_of_Open_Linear_Transformation_between_Normed_Vector_Spaces | https://proofwiki.org/wiki/Characterization_of_Open_Linear_Transformation_between_Normed_Vector_Spaces | [
"Open Mappings",
"Linear Transformations",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Linear Transformation",
"Definition:Closed Unit Ball",
"Definition:Open Mapping"
] | [
"Definition:Open Unit Ball"
] |
proofwiki-20995 | Real Part of Integer Power of Complex Number is Harmonic | Let $z \in \C$ be a complex number.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $z^n$ denote $z$ raised to the $n$th power.
Then the real part $\map \Re {z^n}$ of $z^n$ is a harmonic polynomial. | Let $z = x + i y$.
Then:
{{begin-eqn}}
{{eqn | l = z^n
| r = \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom n j x^{n - j} y^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom n j ... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $z^n$ denote $z$ raised to the [[Definition:Integer Power|$n$th power]].
Then the [[Definition:Real Part|real part]] $\map \Re {z^n}$ of $z^n$ is a [[D... | Let $z = x + i y$.
Then:
{{begin-eqn}}
{{eqn | l = z^n
| r = \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom n j x^{n - j} y^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom n ... | Real Part of Integer Power of Complex Number is Harmonic | https://proofwiki.org/wiki/Real_Part_of_Integer_Power_of_Complex_Number_is_Harmonic | https://proofwiki.org/wiki/Real_Part_of_Integer_Power_of_Complex_Number_is_Harmonic | [
"Real Parts",
"Harmonic Polynomials",
"Complex Powers"
] | [
"Definition:Complex Number",
"Definition:Strictly Positive/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Complex Number/Real Part",
"Definition:Harmonic Polynomial"
] | [
"Arbitrary Power of Complex Number",
"Translation of Index Variable of Summation",
"Symmetry Rule for Binomial Coefficients",
"Factors of Binomial Coefficient",
"Factors of Binomial Coefficient",
"Factors of Binomial Coefficient",
"Factors of Binomial Coefficient",
"Symmetry Rule for Binomial Coeffici... |
proofwiki-20996 | Linear Transformation between Normed Vector Spaces is Open iff Image of Open Unit Ball is Open | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ be a linear transformation.
Let $B_X^O$ be the open unit ball of $X$.
Then $T$ is open {{iff}} $T \sqbrk {B_X^O}$ is open. | === Necessary Condition ===
Suppose that $T$ is open.
From Open Ball is Open Set, $B_X^O$ is open in $\struct {X, \norm {\, \cdot \,}_X}$.
So $T \sqbrk {B_X^O}$ is open in $\struct {Y, \norm {\, \cdot \,}_Y}$.
{{qed|lemma}} | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Let $B_X^O$ be the [[Definition:Open Unit Ball|ope... | === Necessary Condition ===
Suppose that $T$ is [[Definition:Open Mapping|open]].
From [[Open Ball is Open Set]], $B_X^O$ is [[Definition:Open Set|open]] in $\struct {X, \norm {\, \cdot \,}_X}$.
So $T \sqbrk {B_X^O}$ is [[Definition:Open Set|open]] in $\struct {Y, \norm {\, \cdot \,}_Y}$.
{{qed|lemma}} | Linear Transformation between Normed Vector Spaces is Open iff Image of Open Unit Ball is Open | https://proofwiki.org/wiki/Linear_Transformation_between_Normed_Vector_Spaces_is_Open_iff_Image_of_Open_Unit_Ball_is_Open | https://proofwiki.org/wiki/Linear_Transformation_between_Normed_Vector_Spaces_is_Open_iff_Image_of_Open_Unit_Ball_is_Open | [
"Linear Transformations",
"Open Mappings",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Linear Transformation",
"Definition:Open Unit Ball",
"Definition:Open Mapping",
"Definition:Open Set"
] | [
"Definition:Open Mapping",
"Open Ball is Open Set",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Open Set",
"Definition:Open Mapping"
] |
proofwiki-20997 | Norm of Bounded Linear Transformation in terms of Closed Unit Ball | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ be a linear transformation.
Let $B_X^-$ and $B_Y^-$ be the closed unit balls of $X$ and $Y$ respectively.
Let $M > 0$.
Then $T$ is bounded with $\norm T_{... | === Necessary Condition ===
Let $T$ be a bounded linear transformation with $\norm T_{\map B {X, Y} } \le M$.
Then for each $x \in X$ we have:
:$\norm {T x}_Y \le M \norm x_X$
Then if $x \in B_X^-$, we have $\norm x_X \le 1$ and hence:
:$\norm {T x}_Y \le M$
So we have:
:$T \sqbrk {B_X^-} \subseteq M B_Y^-$
{{qed|le... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Let $B_X^-$ and $B_Y^-$ be the [[Definition:Closed U... | === Necessary Condition ===
Let $T$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]] with $\norm T_{\map B {X, Y} } \le M$.
Then for each $x \in X$ we have:
:$\norm {T x}_Y \le M \norm x_X$
Then if $x \in B_X^-$, we have $\norm x_X \le 1$ and hence:
:$\norm {T x}_Y \le M$
So we have... | Norm of Bounded Linear Transformation in terms of Closed Unit Ball | https://proofwiki.org/wiki/Norm_of_Bounded_Linear_Transformation_in_terms_of_Closed_Unit_Ball | https://proofwiki.org/wiki/Norm_of_Bounded_Linear_Transformation_in_terms_of_Closed_Unit_Ball | [
"Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Linear Transformation",
"Definition:Closed Unit Ball",
"Definition:Bounded Linear Transformation"
] | [
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation"
] |
proofwiki-20998 | Norm in terms of Normed Dual Space | 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 of $X$.
Let $B_{X^\ast}^-$ be the closed unit ball of $X^\ast$.
Then:
:$\ds \norm x = \sup_{f \in B_{X^\ast}^-} \cmod {\map f x}$ | From Fundamental Property of Norm on Bounded Linear Functional, we have:
:$\ds \cmod {\map f x} \le \norm x$
for each $x \in X$.
From Existence of Support Functional, there exists $f \in B_{X^\ast}^-$ such that $\map f x = \norm x$.
Hence we conclude:
:$\ds \norm x = \sup_{f \in B_{X^\ast}^-} \cmod {\map f x}$
{{qed... | 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]] of $X$.
Let $B_{X^\ast}^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] of $X^\ast$.
Then:
:$... | From [[Fundamental Property of Norm on Bounded Linear Functional]], we have:
:$\ds \cmod {\map f x} \le \norm x$
for each $x \in X$.
From [[Existence of Support Functional]], there exists $f \in B_{X^\ast}^-$ such that $\map f x = \norm x$.
Hence we conclude:
:$\ds \norm x = \sup_{f \in B_{X^\ast}^-} \cmod {\map f... | Norm in terms of Normed Dual Space | https://proofwiki.org/wiki/Norm_in_terms_of_Normed_Dual_Space | https://proofwiki.org/wiki/Norm_in_terms_of_Normed_Dual_Space | [
"Normed Dual Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Closed Unit Ball"
] | [
"Fundamental Property of Norm on Bounded Linear Functional",
"Existence of Support Functional",
"Category:Normed Dual Spaces"
] |
proofwiki-20999 | Imaginary Part of Integer Power of Complex Number is Harmonic | Let $z \in \C$ be a complex number.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $z^n$ denote $z$ raised to the $n$th power.
Then the imaginary part $\map \Im {z^n}$ of $z^n$ is a harmonic polynomial. | Let $z = x + i y$.
Then:
{{begin-eqn}}
{{eqn | l = z^n
| r = \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom n j x^{n - j} y^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom n j ... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $z^n$ denote $z$ raised to the [[Definition:Integer Power|$n$th power]].
Then the [[Definition:Imaginary Part|imaginary part]] $\map \Im {z^n}$ of $z^n... | Let $z = x + i y$.
Then:
{{begin-eqn}}
{{eqn | l = z^n
| r = \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ even} } } \paren {-1}^{j / 2} \dbinom n j x^{n - j} y^j} + i \paren {\sum_{\substack {0 \mathop \le j \mathop \le n \\ \text {$j$ odd} } } \paren {-1}^{\paren {j - 1} / 2} \dbinom n ... | Imaginary Part of Integer Power of Complex Number is Harmonic | https://proofwiki.org/wiki/Imaginary_Part_of_Integer_Power_of_Complex_Number_is_Harmonic | https://proofwiki.org/wiki/Imaginary_Part_of_Integer_Power_of_Complex_Number_is_Harmonic | [
"Imaginary Parts",
"Harmonic Polynomials",
"Complex Powers"
] | [
"Definition:Complex Number",
"Definition:Strictly Positive/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Complex Number/Imaginary Part",
"Definition:Harmonic Polynomial"
] | [
"Arbitrary Power of Complex Number",
"Translation of Index Variable of Summation",
"Symmetry Rule for Binomial Coefficients",
"Factors of Binomial Coefficient",
"Factors of Binomial Coefficient",
"Factors of Binomial Coefficient",
"Factors of Binomial Coefficient",
"Symmetry Rule for Binomial Coeffici... |
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