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-22900 | Hom Bifunctor With Left Identity Functor is Hom Bifunctor | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$ be a locally small category.
Let $\operatorname{id}_{\mathbf C} : \mathbf C \to \mathbf C$ denote the identity functor.
Let $\map {\operatorname{Hom}_{\mathbf C} } {\operatorname{id}_{\mathbf C}-, -} : \mathbf C^{\text{op} } \times \mathbf C \to \mathbf {Set}... | For each object $\tuple {D^\text{op}, C}$ in $\mathbf C^{\text{op} } \times \mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \map {\operatorname{Hom}_{\mathbf C} } {\operatorname{id}_{\mathbf C} D^\text{op}, C}
| r = \map {\operatorname{Hom}_{\mathbf C} } {\operatorname{id}_{\mathbf C} D, C}
| c = {{Defof|Hom ... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\operatorname{id}_{\mathbf C} : \mathbf C \to \mathbf C$ denote the [[Definition:Identity Functor|identity functor]].
Let $\map {\operatorname{Hom}_{\ma... | For each [[Definition:Object (Category Theory)|object]] $\tuple {D^\text{op}, C}$ in $\mathbf C^{\text{op} } \times \mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \map {\operatorname{Hom}_{\mathbf C} } {\operatorname{id}_{\mathbf C} D^\text{op}, C}
| r = \map {\operatorname{Hom}_{\mathbf C} } {\operatorname{id}_{\... | Hom Bifunctor With Left Identity Functor is Hom Bifunctor | https://proofwiki.org/wiki/Hom_Bifunctor_With_Left_Identity_Functor_is_Hom_Bifunctor | https://proofwiki.org/wiki/Hom_Bifunctor_With_Left_Identity_Functor_is_Hom_Bifunctor | [
"Bifunctors"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Identity Functor",
"Definition:Hom Bifunctor With Left Functor",
"Definition:Hom Bifunctor"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism"
] |
proofwiki-22901 | Hom Bifunctor With Right Identity Functor is Hom Bifunctor | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$ be a locally small category.
Let $\operatorname{id}_{\mathbf C} : \mathbf C \to \mathbf C$ denote the identity functor.
Let $\map {\operatorname{Hom}_{\mathbf C} } {-, \operatorname{id}_{\mathbf C}-} : \mathbf C^{\text{op} } \times \mathbf C \to \mathbf {Set}... | For each object $\tuple {D^\text{op}, C}$ in $\mathbf C^{\text{op} } \times \mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \map {\operatorname{Hom}_{\mathbf C} } {D^\text{op}, \operatorname{id}_{\mathbf C} C}
| r = \map {\operatorname{Hom}_{\mathbf C} } {D, \operatorname{id}_{\mathbf C} C}
| c = {{Defof|Hom ... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\operatorname{id}_{\mathbf C} : \mathbf C \to \mathbf C$ denote the [[Definition:Identity Functor|identity functor]].
Let $\map {\operatorname{Hom}_{\ma... | For each [[Definition:Object (Category Theory)|object]] $\tuple {D^\text{op}, C}$ in $\mathbf C^{\text{op} } \times \mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \map {\operatorname{Hom}_{\mathbf C} } {D^\text{op}, \operatorname{id}_{\mathbf C} C}
| r = \map {\operatorname{Hom}_{\mathbf C} } {D, \operatorname{id}... | Hom Bifunctor With Right Identity Functor is Hom Bifunctor | https://proofwiki.org/wiki/Hom_Bifunctor_With_Right_Identity_Functor_is_Hom_Bifunctor | https://proofwiki.org/wiki/Hom_Bifunctor_With_Right_Identity_Functor_is_Hom_Bifunctor | [
"Bifunctors"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Identity Functor",
"Definition:Hom Bifunctor With Right Functor",
"Definition:Hom Bifunctor"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism"
] |
proofwiki-22902 | Curve with Constant Zero Curvature is Straight Line | Let $\CC$ be a curve whose curvature is constant and zero.
Then $\CC$ is a straight line. | From Straight Line has Zero Curvature, if $\CC$ is a straight line then its curvature is indeed constant and zero.
It remains to be demonstrated that every such curve with constant zero curvature is a straight line.
{{ProofWanted}} | Let $\CC$ be a [[Definition:Curve|curve]] whose [[Definition:Curvature|curvature]] is [[Definition:Constant|constant]] and [[Definition:Zero (Number)|zero]].
Then $\CC$ is a [[Definition:Straight Line|straight line]]. | From [[Straight Line has Zero Curvature]], if $\CC$ is a [[Definition:Straight Line|straight line]] then its [[Definition:Curvature|curvature]] is indeed [[Definition:Constant|constant]] and [[Definition:Zero (Number)|zero]].
It remains to be demonstrated that every such [[Definition:Curve|curve]] with [[Definition:Co... | Curve with Constant Zero Curvature is Straight Line | https://proofwiki.org/wiki/Curve_with_Constant_Zero_Curvature_is_Straight_Line | https://proofwiki.org/wiki/Curve_with_Constant_Zero_Curvature_is_Straight_Line | [
"Curvature",
"Straight Lines"
] | [
"Definition:Line/Curve",
"Definition:Curvature",
"Definition:Constant",
"Definition:Zero (Number)",
"Definition:Line/Straight Line"
] | [
"Straight Line has Zero Curvature",
"Definition:Line/Straight Line",
"Definition:Curvature",
"Definition:Constant",
"Definition:Zero (Number)",
"Definition:Line/Curve",
"Definition:Constant",
"Definition:Zero (Number)",
"Definition:Curvature",
"Definition:Line/Straight Line"
] |
proofwiki-22903 | Open Real Interval is not Homeomorphic to Half-Open Real Interval | Let $I_o$ denote the open real interval $\openint 0 1$.
Let $I_h$ denote the half-open real interval $\hointl 0 1$.
Then $I_o$ and $I_h$ are not homeomorphic. | From Every Point except Endpoint in Connected Linearly Ordered Space is Cut Point:
:every point of $I_o$ is a cut point
:not every point of $I_h$ is a cut point.
Indeed, the point $1$ is not a cut point of $I_h$, as:
:$I_h \setminus \set 1 = I_o$
which is a connected space.
{{qed}} | Let $I_o$ denote the [[Definition:Open Real Interval|open real interval]] $\openint 0 1$.
Let $I_h$ denote the [[Definition:Half-Open Real Interval|half-open real interval]] $\hointl 0 1$.
Then $I_o$ and $I_h$ are not [[Definition:Homeomorphic Topological Spaces|homeomorphic]]. | From [[Every Point except Endpoint in Connected Linearly Ordered Space is Cut Point]]:
:every [[Definition:Point|point]] of $I_o$ is a [[Definition:Cut Point|cut point]]
:not every [[Definition:Point|point]] of $I_h$ is a [[Definition:Cut Point|cut point]].
Indeed, the [[Definition:Point|point]] $1$ is not a [[Definit... | Open Real Interval is not Homeomorphic to Half-Open Real Interval | https://proofwiki.org/wiki/Open_Real_Interval_is_not_Homeomorphic_to_Half-Open_Real_Interval | https://proofwiki.org/wiki/Open_Real_Interval_is_not_Homeomorphic_to_Half-Open_Real_Interval | [
"Homeomorphisms",
"Real Intervals"
] | [
"Definition:Real Interval/Open",
"Definition:Real Interval/Half-Open",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Every Point except Endpoint in Connected Linearly Ordered Space is Cut Point",
"Definition:Point",
"Definition:Cut Point",
"Definition:Point",
"Definition:Cut Point",
"Definition:Point",
"Definition:Cut Point",
"Definition:Connected Topological Space"
] |
proofwiki-22904 | Comma Category is Category | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be categories.
Let $F : \mathbf D \to \mathbf C$ and $G : \mathbf E \to \mathbf C$ be covariant functors.
Let $\paren{G \downarrow F}$ denote the comma category $G$ over $F$.
Then:
:$\paren{G \downarrow F}$ is a metacategory. | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a metacategory. | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F : \mathbf D \to \mathbf C$ and $G : \mathbf E \to \mathbf C$ be [[Definition:Covariant Functor|covariant functors]].
Let $\paren{G \downarrow F}$ denote the [[Definition:Comma Category|comma category $G$ over $F$]].
Then:
:... | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a [[Definition:Metacategory|metacategory]]. | Comma Category is Category | https://proofwiki.org/wiki/Comma_Category_is_Category | https://proofwiki.org/wiki/Comma_Category_is_Category | [
"Comma Categories"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Comma Category",
"Definition:Metacategory"
] | [
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory"
] |
proofwiki-22905 | Functor Under Object Comma Category is Category | Let $\mathbf C$, $\mathbf D$ be categories.
Let $C$ be an object of $\mathbf C$.
Let $F : \mathbf D \to \mathbf C$ be a covariant functor.
Let $\paren{C \downarrow F}$ denote the comma category $F$ under $C$.
Then:
:$\paren{C \downarrow F}$ is a metacategory. | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a metacategory. | Let $\mathbf C$, $\mathbf D$ be [[Definition:Category|categories]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $F : \mathbf D \to \mathbf C$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\paren{C \downarrow F}$ denote the [[Definition:Functor Under Object Comm... | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a [[Definition:Metacategory|metacategory]]. | Functor Under Object Comma Category is Category | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Category | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Category | [
"Comma Categories"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Comma Category/Functor Under Object",
"Definition:Metacategory"
] | [
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory"
] |
proofwiki-22906 | Functor Over Object Comma Category is Category | Let $\mathbf C$, $\mathbf E$ be categories.
Let $C$ be an object of $\mathbf C$.
Let $G : \mathbf E \to \mathbf C$ be a covariant functor.
Let $\paren{G \downarrow C}$ denote the comma category $G$ over $C$.
Then:
:$\paren{G \downarrow C}$ is a metacategory | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a metacategory. | Let $\mathbf C$, $\mathbf E$ be [[Definition:Category|categories]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $G : \mathbf E \to \mathbf C$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\paren{G \downarrow C}$ denote the [[Definition:Functor Over Object Comma... | Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a [[Definition:Metacategory|metacategory]]. | Functor Over Object Comma Category is Category | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Category | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Category | [
"Comma Categories"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Comma Category/Functor Over Object",
"Definition:Metacategory"
] | [
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Metacategory"
] |
proofwiki-22907 | Functor Under Object Comma Category is Isomorphic to Comma Category | Let $\mathbf C$, $\mathbf D$ be categories.
Let $C$ be an object of $\mathbf C$.
Let $F : \mathbf D \to \mathbf C$ be a covariant functor.
Let $\mathbf 1$ denote the category one.
Let $G: \mathbf 1\to \mathbf C$ be the functor defined by:
{{DefineFunctor
|ob = $G* = C$
|mor = $G \operatorname{id}_* = \operatorname{id}_... | === Lemma 1 ===
{{:Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 1}}{{qed|lemma}} | Let $\mathbf C$, $\mathbf D$ be [[Definition:Category|categories]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $F : \mathbf D \to \mathbf C$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\mathbf 1$ denote the [[Definition:One (Category)|category one]].
Let $G:... | === [[Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 1|Lemma 1]] ===
{{:Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 1}}{{qed|lemma}} | Functor Under Object Comma Category is Isomorphic to Comma Category | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Isomorphic_to_Comma_Category | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Isomorphic_to_Comma_Category | [
"Comma Categories",
"Functor Under Object Comma Category is Isomorphic to Comma Category"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:One (Category)",
"Definition:Functor/Covariant",
"Definition:Comma Category/Functor Under Object",
"Definition:Comma Category/General Form",
"Definition:Isomorphism of Categories",
"Definition:O... | [
"Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 1"
] |
proofwiki-22908 | Functor Over Object Comma Category is Isomorphic to Comma Category | Let $\mathbf C$, $\mathbf D$ be categories.
Let $C$ be an object of $\mathbf C$.
Let $G : \mathbf D \to \mathbf C$ be a covariant functor.
Let $\mathbf 1$ denote the category one.
Let $F: \mathbf 1\to \mathbf C$ be the functor defined by:
{{DefineFunctor
|ob = $F* = C$
|mor = $F \operatorname{id}_* = \operatorname{id}_... | === Lemma 1===
{{:Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 1}}{{qed|lemma}} | Let $\mathbf C$, $\mathbf D$ be [[Definition:Category|categories]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $G : \mathbf D \to \mathbf C$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\mathbf 1$ denote the [[Definition:One (Category)|category one]].
Let $F:... | === [[Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 1|Lemma 1]]===
{{:Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 1}}{{qed|lemma}} | Functor Over Object Comma Category is Isomorphic to Comma Category | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Isomorphic_to_Comma_Category | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Isomorphic_to_Comma_Category | [
"Comma Categories",
"Functor Over Object Comma Category is Isomorphic to Comma Category"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:One (Category)",
"Definition:Functor/Covariant",
"Definition:Comma Category/Functor Over Object",
"Definition:Comma Category/General Form",
"Definition:Isomorphism of Categories",
"Definition:Ob... | [
"Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 1"
] |
proofwiki-22909 | Coslice Category is Isomorphic to Comma Category | Let $\mathbf C$ be a category.
Let $C$ be an object of $\mathbf C$.
Let $\operatorname{id}_{\mathbf C}$ denote the identity functor on $\mathbf C$.
Let $\mathbf 1$ denote the category one.
Let $G: \mathbf 1\to \mathbf C$ be the functor defined by:
:$G* = C$
and
:$G \operatorname{id}_* = \operatorname{id}_C$
Let $C / \m... | === Lemma 1 ===
{{:Coslice Category is Isomorphic to Comma Category/Lemma 1}}{{qed|lemma}} | Let $\mathbf C$ be a [[Definition:Category|category]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $\operatorname{id}_{\mathbf C}$ denote the [[Definition:Identity Functor|identity functor]] on $\mathbf C$.
Let $\mathbf 1$ denote the [[Definition:One (Category)|category one]].
... | === [[Coslice Category is Isomorphic to Comma Category/Lemma 1|Lemma 1]] ===
{{:Coslice Category is Isomorphic to Comma Category/Lemma 1}}{{qed|lemma}} | Coslice Category is Isomorphic to Comma Category | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Comma_Category | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Comma_Category | [
"Coslice Category is Isomorphic to Comma Category",
"Coslice Categories",
"Comma Categories"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Identity Functor",
"Definition:One (Category)",
"Definition:Functor/Covariant",
"Definition:Coslice Category",
"Definition:Comma Category/General Form",
"Definition:Isomorphism of Categories",
"Definition:Object (Category Theo... | [
"Coslice Category is Isomorphic to Comma Category/Lemma 1"
] |
proofwiki-22910 | Slice Category is Isomorphic to Comma Category | Let $\mathbf C$ be a category.
Let $C$ be an object of $\mathbf C$.
Let $\operatorname{id}_{\mathbf C}$ denote the identity functor on $\mathbf C$.
Let $\mathbf 1$ denote the category one.
Let $F: \mathbf 1\to \mathbf C$ be the functor defined by:
:$F* = C$
and
:$F \operatorname{id}_* = \operatorname{id}_C$
Let $\mathb... | === Lemma 1 ===
{{:Slice Category is Isomorphic to Comma Category/Lemma 1}}{{qed|lemma}} | Let $\mathbf C$ be a [[Definition:Category|category]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $\operatorname{id}_{\mathbf C}$ denote the [[Definition:Identity Functor|identity functor]] on $\mathbf C$.
Let $\mathbf 1$ denote the [[Definition:One (Category)|category one]].
... | === [[Slice Category is Isomorphic to Comma Category/Lemma 1|Lemma 1]] ===
{{:Slice Category is Isomorphic to Comma Category/Lemma 1}}{{qed|lemma}} | Slice Category is Isomorphic to Comma Category | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Comma_Category | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Comma_Category | [
"Slice Categories",
"Comma Categories",
"Slice Category is Isomorphic to Comma Category"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Identity Functor",
"Definition:One (Category)",
"Definition:Functor/Covariant",
"Definition:Slice Category",
"Definition:Comma Category/General Form",
"Definition:Isomorphism of Categories",
"Definition:Object (Category Theory... | [
"Slice Category is Isomorphic to Comma Category/Lemma 1"
] |
proofwiki-22911 | Morphism Category is Isomorphic to Comma Category | Let $\mathbf C$ be a category.
Let $C$ be an object of $\mathbf C$.
Let $\operatorname{id}_{\mathbf C}$ denote the identity functor on $\mathbf C$.
Let $\mathbf C^\to$ denote the morphism category on $\mathbf C$.
Let $\paren{\operatorname{id}_{\mathbf C} \downarrow \operatorname{id}_{\mathbf C}}$ denote the comma categ... | === Lemma 1 ===
{{:Morphism Category is Isomorphic to Comma Category/Lemma 1}}{{qed|lemma}} | Let $\mathbf C$ be a [[Definition:Category|category]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $\operatorname{id}_{\mathbf C}$ denote the [[Definition:Identity Functor|identity functor]] on $\mathbf C$.
Let $\mathbf C^\to$ denote the [[Definition:Morphism Category|morphism ... | === [[Morphism Category is Isomorphic to Comma Category/Lemma 1|Lemma 1]] ===
{{:Morphism Category is Isomorphic to Comma Category/Lemma 1}}{{qed|lemma}} | Morphism Category is Isomorphic to Comma Category | https://proofwiki.org/wiki/Morphism_Category_is_Isomorphic_to_Comma_Category | https://proofwiki.org/wiki/Morphism_Category_is_Isomorphic_to_Comma_Category | [
"Morphism Categories",
"Comma Categories",
"Morphism Category is Isomorphic to Comma Category"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Identity Functor",
"Definition:Morphism Category",
"Definition:Comma Category/General Form",
"Definition:Isomorphism of Categories",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Th... | [
"Morphism Category is Isomorphic to Comma Category/Lemma 1"
] |
proofwiki-22912 | Equivalence of Definitions of Universal Morphism from Object to Functor | Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $C$ be an object of $\mathbf C$.
Let $F: \mathbf D \to \mathbf C$ be a covariant functor.
Let the pair $\tuple{R, u}$ consist of an object $R$ of $\mathbf D$ and a morphism $u: C \to FR$ in $C$.
{{TFAE|def=Universal Morphism from Object to Functor|context=Category ... | The result follows immediately from:
* Definition:Initial Object
* Definition:Functor Under Object Comma Category
{{qed}} | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Metacategory|metacategories]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $F: \mathbf D \to \mathbf C$ be a [[Definition:Covariant Functor|covariant functor]].
Let the [[Definition:Ordered Pair|pair]] $\tuple{R, u}$ consist of an ... | The result follows immediately from:
* [[Definition:Initial Object]]
* [[Definition:Functor Under Object Comma Category]]
{{qed}} | Equivalence of Definitions of Universal Morphism from Object to Functor | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Universal_Morphism_from_Object_to_Functor | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Universal_Morphism_from_Object_to_Functor | [
"Universal Morphisms",
"Comma Categories"
] | [
"Definition:Metacategory",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Ordered Pair",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Object to Functor/Definition 1",
"Definition:Universal Morphism from Object to ... | [
"Definition:Initial Object",
"Definition:Comma Category/Functor Under Object"
] |
proofwiki-22913 | Morphism of Unit of Adjunction is Universal | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $\tuple {F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
Let $\eta: \operatorname {id}_{\mathbf D} \to GF$ be the unit of adjunction $\tuple {F, G, \alpha}$
Then:
:for each object $D$ in $\mathbf D$ the morphism $\eta_D$ is a universal ... | By definition of unit of adjunction:
:* for each $D$ of $\mathbf D: \eta_{_D} = \map {\alpha_{\tuple{D, F D}}} {\operatorname {id}_{FD}}$ where
::* $\alpha_{\tuple{D, F D}} : \map {\operatorname{Hom}_{\mathbf C} } {F D, F D} \to \map {\operatorname{Hom}_{\mathbf D} } {D, GF D}$ is the bijection from the adjunction $\tu... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $\tuple {F, G, \alpha}$ be an [[Definition:Adjunction|adjunction]] between $\mathbf C$ and $\mathbf D$.
Let $\eta: \operatorname {id}_{\mathbf D} \to GF$ be the [[Definition:Unit of Adjunction|unit of adjunction $\tup... | By definition of [[Definition:Unit of Adjunction|unit of adjunction]]:
:* for each $D$ of $\mathbf D: \eta_{_D} = \map {\alpha_{\tuple{D, F D}}} {\operatorname {id}_{FD}}$ where
::* $\alpha_{\tuple{D, F D}} : \map {\operatorname{Hom}_{\mathbf C} } {F D, F D} \to \map {\operatorname{Hom}_{\mathbf D} } {D, GF D}$ is the ... | Morphism of Unit of Adjunction is Universal | https://proofwiki.org/wiki/Morphism_of_Unit_of_Adjunction_is_Universal | https://proofwiki.org/wiki/Morphism_of_Unit_of_Adjunction_is_Universal | [
"Adjunctions",
"Universal Morphisms"
] | [
"Definition:Locally Small Category",
"Definition:Adjunction",
"Definition:Unit of Adjunction",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Object to Functor"
] | [
"Definition:Unit of Adjunction",
"Definition:Bijection",
"Definition:Adjunction",
"Definition:Identity Morphism",
"Definition:Identity Functor",
"Definition:Composition of Functors",
"Definition:Universal Morphism from Object to Functor",
"Definition:Morphism",
"Definition:Unique",
"Definition:Mor... |
proofwiki-22914 | Equivalence of Definitions of Universal Morphism from Functor to Object | Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $C$ be an object of $\mathbf C$.
Let $F: \mathbf D \to \mathbf C$ be a covariant functor.
Let the pair $\tuple{R, v}$ consist of an object $R$ of $\mathbf D$ and a morphism $v: FR \to C$ in $C$.
{{TFAE|def=Universal Morphism from Functor to Object|context=Category ... | The result follows immediately from:
* Definition:Terminal Object
* Definition:Functor Over Object Comma Category
{{qed}} | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Metacategory|metacategories]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $F: \mathbf D \to \mathbf C$ be a [[Definition:Covariant Functor|covariant functor]].
Let the [[Definition:Ordered Pair|pair]] $\tuple{R, v}$ consist of an ... | The result follows immediately from:
* [[Definition:Terminal Object]]
* [[Definition:Functor Over Object Comma Category]]
{{qed}} | Equivalence of Definitions of Universal Morphism from Functor to Object | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Universal_Morphism_from_Functor_to_Object | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Universal_Morphism_from_Functor_to_Object | [
"Universal Morphisms",
"Comma Categories"
] | [
"Definition:Metacategory",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Ordered Pair",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Functor to Object/Definition 1",
"Definition:Universal Morphism from Functor to... | [
"Definition:Terminal Object",
"Definition:Comma Category/Functor Over Object"
] |
proofwiki-22915 | Morphism of Counit of Adjunction is Universal | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $\tuple {F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
Let $\xi: FG \to \operatorname {id}_{\mathbf C}$ be the counit of adjunction $\tuple {F, G, \alpha}$
Then:
:for each object $C$ in $\mathbf C$ the morphism $\xi_C$ is a universal ... | Let $\beta$ be the inverse of the natural isomorphism $\alpha$.
By definition of counit of adjunction:
:* for each $C$ of $\mathbf C: \xi_{C} = \map {\beta_{\tuple{GC, C}}} {\operatorname {id}_{GC}}$ where
::* $\beta_{\tuple{GC, C} }: \map {\operatorname{Hom}_{\mathbf C} } {F G C, C} \to \map {\operatorname{Hom}_{\math... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $\tuple {F, G, \alpha}$ be an [[Definition:Adjunction|adjunction]] between $\mathbf C$ and $\mathbf D$.
Let $\xi: FG \to \operatorname {id}_{\mathbf C}$ be the [[Definition:Counit of Adjunction|counit of adjunction $\... | Let $\beta$ be the [[Definition:Inverse Natural Isomorphism between Covariant Functors|inverse]] of the [[Definition:Natural Isomorphism|natural isomorphism]] $\alpha$.
By definition of [[Definition:Counit of Adjunction|counit of adjunction]]:
:* for each $C$ of $\mathbf C: \xi_{C} = \map {\beta_{\tuple{GC, C}}} {\op... | Morphism of Counit of Adjunction is Universal | https://proofwiki.org/wiki/Morphism_of_Counit_of_Adjunction_is_Universal | https://proofwiki.org/wiki/Morphism_of_Counit_of_Adjunction_is_Universal | [
"Adjunctions",
"Universal Morphisms"
] | [
"Definition:Locally Small Category",
"Definition:Adjunction",
"Definition:Counit of Adjunction",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Functor to Object"
] | [
"Definition:Natural Isomorphism between Covariant Functors/Inverse",
"Definition:Natural Isomorphism",
"Definition:Counit of Adjunction",
"Definition:Bijection",
"Definition:Natural Isomorphism between Covariant Functors/Inverse",
"Definition:Identity Morphism",
"Definition:Identity Functor",
"Definit... |
proofwiki-22916 | Unit of Adjunction Induces Adjunction | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be covariant functors.
Let $\eta: \operatorname {id}_{\mathbf D} \to GF$ be a natural transformation such that:
:for each object $D$ in $\mathbf D$ the morphism $\eta_D$ is a universal morphism fr... | From Characterization of Adjunction Using Right Adjuncts of Morphisms:
::the triple $\tuple {F, G, \alpha}$ is an adjunction
{{iff}}:
:$(1)\quad$ For each $D$ in $\mathbf D$ and $C$ in $\mathbf C : \alpha_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf C} } {FD, C} \to \map {\mathrm {Hom}_{\mathbf D} } {D, GC}$ is a bijec... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta: \operatorname {id}_{\mathbf D} \to GF$ be a [[Definition:Natural Transformation|natural t... | From [[Characterization of Adjunction Using Right Adjuncts of Morphisms]]:
::the [[Definition:Triple|triple]] $\tuple {F, G, \alpha}$ is an [[Definition:Adjunction|adjunction]]
{{iff}}:
:$(1)\quad$ For each $D$ in $\mathbf D$ and $C$ in $\mathbf C : \alpha_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf C} } {FD, C} \to \... | Unit of Adjunction Induces Adjunction | https://proofwiki.org/wiki/Unit_of_Adjunction_Induces_Adjunction | https://proofwiki.org/wiki/Unit_of_Adjunction_Induces_Adjunction | [
"Adjunctions",
"Universal Morphisms"
] | [
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Object to Functor",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Defi... | [
"Characterization of Adjunction Using Right Adjuncts of Morphisms",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction",
"Definition:Bijection",
"Definition:Bijection",
"Characterization of Adjunction Using Right Adjuncts of Morphisms",
"Definition:Adjunction"
] |
proofwiki-22917 | Counit of Adjunction Induces Adjunction | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be functors.
Let $\xi: FG \to \operatorname {id}_{\mathbf C}$ be a natural transformation such that:
:for each object $C$ in $\mathbf C$ the morphism $\xi_C$ is a universal morphism from $F$ to $G... | From Characterization of Adjunction Using Left Adjuncts of Morphisms, we have:
:$(1)\quad$For each $D$ in $\mathbf D$ and $C$ in $\mathbf C : \beta_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf D} } {D, GC} \to \map {\mathrm {Hom}_{\mathbf C} } {FD, C}$ is a bijection
:$(2):\quad$ for every $g:C_1 \to C_2 \in \mathbf C$... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be [[Definition:Functor|functors]].
Let $\xi: FG \to \operatorname {id}_{\mathbf C}$ be a [[Definition:Natural Transformation|natural transformation]] such ... | From [[Characterization of Adjunction Using Left Adjuncts of Morphisms]], we have:
:$(1)\quad$For each $D$ in $\mathbf D$ and $C$ in $\mathbf C : \beta_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf D} } {D, GC} \to \map {\mathrm {Hom}_{\mathbf C} } {FD, C}$ is a [[Definition:Bijection|bijection]]
:$(2):\quad$ for ever... | Counit of Adjunction Induces Adjunction | https://proofwiki.org/wiki/Counit_of_Adjunction_Induces_Adjunction | https://proofwiki.org/wiki/Counit_of_Adjunction_Induces_Adjunction | [
"Adjunctions",
"Universal Morphisms"
] | [
"Definition:Locally Small Category",
"Definition:Functor",
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Functor to Object",
"Definition:Object (Category Theory)",
"Definition:Mapping",
"Definition:Inverse Mappi... | [
"Characterization of Adjunction Using Left Adjuncts of Morphisms",
"Definition:Bijection",
"Definition:Bijection",
"Characterization of Adjunction Using Left Adjuncts of Morphisms"
] |
proofwiki-22918 | Convex Hull of Union of Compact Set and Closed von Neumann-Bounded Set in Hausdorff Topological Vector Space is Closed | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a Hausdorff topological vector space.
Let $A \subseteq X$ be compact.
Let $B \subseteq X$ be closed and von Neumann-bounded.
Let $C = \map {\mathrm {conv} } {A \cup B}$ be the convex hull of $A \cup B$.
Then $C$ is closed. | We show that every convergent net in $C$ has its limit in $C$.
Let $\struct {\AA, \preceq_{\AA} }$ be a directed set.
Let $\family {z_\alpha}_{\alpha \mathop \in \AA}$ be a net in $C$ converging to $z$.
We want to show that $z \in C$.
From Convex Hull of Finite Union of Convex Sets, there exists nets:
:$\family {x_\al... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Hausdorff Topological Vector Space|Hausdorff topological vector space]].
Let $A \subseteq X$ be [[Definition:Compact Topological Space|compact]].
Let $B \subseteq X$ be [[Definition:Closed Set|closed]] and [[Definition:von Neumann-Bounded Subset ... | We show that every [[Definition:Convergent Net|convergent net]] in $C$ has its [[Definition:Limit of Net|limit]] in $C$.
Let $\struct {\AA, \preceq_{\AA} }$ be a [[Definition:Directed Set|directed set]].
Let $\family {z_\alpha}_{\alpha \mathop \in \AA}$ be a [[Definition:Net (Set Theory)|net]] in $C$ [[Definition:Con... | Convex Hull of Union of Compact Set and Closed von Neumann-Bounded Set in Hausdorff Topological Vector Space is Closed | https://proofwiki.org/wiki/Convex_Hull_of_Union_of_Compact_Set_and_Closed_von_Neumann-Bounded_Set_in_Hausdorff_Topological_Vector_Space_is_Closed | https://proofwiki.org/wiki/Convex_Hull_of_Union_of_Compact_Set_and_Closed_von_Neumann-Bounded_Set_in_Hausdorff_Topological_Vector_Space_is_Closed | [
"von Neumann-Bounded Subsets of Topological Vector Spaces",
"Convex Hulls",
"Compact Topological Spaces"
] | [
"Definition:Hausdorff Topological Vector Space",
"Definition:Compact Topological Space",
"Definition:Closed Set",
"Definition:von Neumann-Bounded Subset of Topological Vector Space",
"Definition:Convex Hull",
"Definition:Closed Set"
] | [
"Definition:Convergent Net",
"Definition:Limit of Net",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Convex Hull of Finite Union of Convex Sets",
"Definition:Net (Set Theory)",
"Definition:Compact Topological Space",
"Definition:Directed Preordering... |
proofwiki-22919 | Characterization of Convergence of Net in Initial Topology | Let $X$ be a set.
Let $\FF$ be a set of functions $f : X \to Y_f$.
For each $f \in \FF$, let $\tau_f$ be a topology on $Y_f$.
Let $\tau$ be the initial topology on $X$ induced by $\FF$ with $f \in \FF$ given the topology $\tau_f$.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {x_\lambda}_{\lambda ... | === Necessary Condition ===
Suppose that $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ converges to $x$.
Let $f \in \FF$.
From the definition of the initial topology, $f$ is continuous.
From Characterization of Continuity in terms of Nets, we have:
:$\map f {x_\lambda} \to \map f x$ in $\struct {Y_f, \tau_f}$.
... | Let $X$ be a [[Definition:Set|set]].
Let $\FF$ be a [[Definition:Set|set]] of [[Definition:Function|functions]] $f : X \to Y_f$.
For each $f \in \FF$, let $\tau_f$ be a [[Definition:Topology|topology]] on $Y_f$.
Let $\tau$ be the [[Definition:Initial Topology|initial topology]] on $X$ induced by $\FF$ with $f \in ... | === Necessary Condition ===
Suppose that $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ [[Definition:Convergent Net|converges]] to $x$.
Let $f \in \FF$.
From the definition of the [[Definition:Initial Topology|initial topology]], $f$ is [[Definition:Continuous Mapping|continuous]].
From [[Characterization of ... | Characterization of Convergence of Net in Initial Topology | https://proofwiki.org/wiki/Characterization_of_Convergence_of_Net_in_Initial_Topology | https://proofwiki.org/wiki/Characterization_of_Convergence_of_Net_in_Initial_Topology | [
"Initial Topology",
"Nets (Set Theory)"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Function",
"Definition:Topology",
"Definition:Initial Topology",
"Definition:Topology",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net"
] | [
"Definition:Convergent Net",
"Definition:Initial Topology",
"Definition:Continuous Mapping",
"Characterization of Continuity in terms of Nets",
"Definition:Convergent Net"
] |
proofwiki-22920 | Characterization of Convergence of Net in Product Topology | Let $I$ be a set.
Let $\family {\tuple {X_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an $I$-indexed family of sets.
Let:
:$\ds X = \prod_{\alpha \mathop \in I} X_\alpha$
Let $\family {\pi_\alpha}_{\alpha \mathop \in I}$ be the coordinate projections $X \to X_\alpha$.
Let $\tau$ be the product topology on $X$.... | Note that $\tau$ is the initial topology on $X$ induced by $\FF = \family {\pi_\alpha}_{\alpha \mathop \in I}$.
The result then follows from Characterization of Convergence of Net in Initial Topology.
{{qed}}
Category:Product Topology
Category:Nets (Set Theory)
fkznrsyme589w5l9sr43d7ui72mhayq | Let $I$ be a [[Definition:Set|set]].
Let $\family {\tuple {X_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family of Sets|$I$-indexed family of sets]].
Let:
:$\ds X = \prod_{\alpha \mathop \in I} X_\alpha$
Let $\family {\pi_\alpha}_{\alpha \mathop \in I}$ be the [[Definition:Projection ... | Note that $\tau$ is the [[Definition:Initial Topology|initial topology]] on $X$ induced by $\FF = \family {\pi_\alpha}_{\alpha \mathop \in I}$.
The result then follows from [[Characterization of Convergence of Net in Initial Topology]].
{{qed}}
[[Category:Product Topology]]
[[Category:Nets (Set Theory)]]
fkznrsyme589... | Characterization of Convergence of Net in Product Topology | https://proofwiki.org/wiki/Characterization_of_Convergence_of_Net_in_Product_Topology | https://proofwiki.org/wiki/Characterization_of_Convergence_of_Net_in_Product_Topology | [
"Nets (Set Theory)",
"Product Topology",
"Nets (Set Theory)"
] | [
"Definition:Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Product Topology",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net"
] | [
"Definition:Initial Topology",
"Characterization of Convergence of Net in Initial Topology",
"Category:Product Topology",
"Category:Nets (Set Theory)"
] |
proofwiki-22921 | Closed Graph Theorem/Compact Hausdorff Codomain | Let $\struct {X, \tau_X}$ be a topological space.
Let $\struct {Y, \tau_Y}$ be a compact $T_2$ (Hausdorff) space.
Equip $X \times Y$ with the product topology $\tau_{X \times Y}$.
Let $f : X \to Y$ be a mapping.
Let $G_f$ be the graph of $f$.
Then $f$ is continuous {{iff}} $G_f$ is closed in $\struct {X \times Y, \tau... | === Necessary Condition ===
Suppose that $f$ is continuous.
From Characterization of Closedness in terms of Nets, it is enough to show that:
:every convergent net in $G_f$ has its limit in $G_f$.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {\tuple {x_\lambda, y_\lambda} }_{\lambda \mathop \in \Lamb... | Let $\struct {X, \tau_X}$ be a [[Definition:Topological Space|topological space]].
Let $\struct {Y, \tau_Y}$ be a [[Definition:Compact Topological Space|compact]] [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Equip $X \times Y$ with the [[Definition:Product Topology|product topology]] $\tau_{X \times Y}$.
Let $f... | === Necessary Condition ===
Suppose that $f$ is [[Definition:Continuous Mapping|continuous]].
From [[Characterization of Closedness in terms of Nets]], it is enough to show that:
:every [[Definition:Convergent Net|convergent net]] in $G_f$ has its [[Definition:Limit of Net|limit]] in $G_f$.
Let $\struct {\Lambda, \p... | Closed Graph Theorem/Compact Hausdorff Codomain | https://proofwiki.org/wiki/Closed_Graph_Theorem/Compact_Hausdorff_Codomain | https://proofwiki.org/wiki/Closed_Graph_Theorem/Compact_Hausdorff_Codomain | [
"Closed Graph Theorem"
] | [
"Definition:Topological Space",
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Product Topology",
"Definition:Mapping",
"Definition:Graph of Mapping",
"Definition:Continuous Mapping",
"Definition:Closed Set"
] | [
"Definition:Continuous Mapping",
"Characterization of Closedness in terms of Nets",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Definition:Directed Preordering",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Characterization of Convergence of Net in Product Topology",
"Defini... |
proofwiki-22922 | Vector Space with Subspace of Algebraic Dual Separating Points is Dual System | Let $K$ be a field.
Let $E$ be a vector space over $K$.
Let $E^\#$ be the algebraic dual of $E$.
Let $F \subseteq E^\#$ be a vector subspace of $E^\#$ that separates points.
Define $\innerprod \cdot \cdot : E \times F \to K$ by:
:$\innerprod x f = \map f x$ for each $\tuple {x, f} \in E \times F$.
Then $\tuple {E, F, ... | We first show that $\innerprod \cdot \cdot$ is a bilinear mapping.
Let $x_1, x_2, x \in E$ and $f_1, f_2, f \in F$ and $\lambda \in K$.
We have:
{{begin-eqn}}
{{eqn | l = \innerprod {x_1 + \lambda x_2} f
| r = \map f {x_1 + \lambda x_2}
}}
{{eqn | r = \map f {x_1} + \lambda \map f {x_2}
| c = {{Defof|Linear Functi... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $E$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $E^\#$ be the [[Definition:Vector Algebraic Dual|algebraic dual]] of $E$.
Let $F \subseteq E^\#$ be a [[Definition:Vector Subspace|vector subspace]] of $E^\#$ that [[Definition:Mappings Sep... | We first show that $\innerprod \cdot \cdot$ is a [[Definition:Bilinear Mapping|bilinear mapping]].
Let $x_1, x_2, x \in E$ and $f_1, f_2, f \in F$ and $\lambda \in K$.
We have:
{{begin-eqn}}
{{eqn | l = \innerprod {x_1 + \lambda x_2} f
| r = \map f {x_1 + \lambda x_2}
}}
{{eqn | r = \map f {x_1} + \lambda \map f {... | Vector Space with Subspace of Algebraic Dual Separating Points is Dual System | https://proofwiki.org/wiki/Vector_Space_with_Subspace_of_Algebraic_Dual_Separating_Points_is_Dual_System | https://proofwiki.org/wiki/Vector_Space_with_Subspace_of_Algebraic_Dual_Separating_Points_is_Dual_System | [
"Dual Systems",
"Vector Space with Subspace of Algebraic Dual Separating Points is Dual System",
"Algebraic Duals"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Algebraic Dual/Vector Space",
"Definition:Vector Subspace",
"Definition:Mappings Separating Points",
"Definition:Dual System"
] | [
"Definition:Bilinear Mapping",
"Definition:Bilinear Mapping",
"Definition:Mappings Separating Points",
"Category:Dual Systems",
"Category:Vector Space with Subspace of Algebraic Dual Separating Points is Dual System",
"Category:Algebraic Duals"
] |
proofwiki-22923 | Algebraic Dual Separates Points of Vector Space | Let $K$ be a field.
Let $E$ be a vector space over $K$.
Let $E^\#$ be the algebraic dual of $E$.
Then $E^\#$ separates points. | We have that $E^\#$ separates points {{iff}}:
:for all $x, y \in E$ with $x \ne y$, there exists $f \in E^\#$ with $\map f x \ne \map f y$.
If $E = \set { {\mathbf 0}_E}$ (that is, $\dim E = 0$), then there are no $x, y \in E$ with $x \ne y$, hence this holds trivially.
Now suppose that $\dim E \ge 1$.
Since each $f \... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $E$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $E^\#$ be the [[Definition:Vector Algebraic Dual|algebraic dual]] of $E$.
Then $E^\#$ [[Definition:Mappings Separating Points|separates points]]. | We have that $E^\#$ [[Definition:Mappings Separating Points|separates points]] {{iff}}:
:for all $x, y \in E$ with $x \ne y$, there exists $f \in E^\#$ with $\map f x \ne \map f y$.
If $E = \set { {\mathbf 0}_E}$ (that is, $\dim E = 0$), then there are no $x, y \in E$ with $x \ne y$, hence this holds trivially.
Now ... | Algebraic Dual Separates Points of Vector Space | https://proofwiki.org/wiki/Algebraic_Dual_Separates_Points_of_Vector_Space | https://proofwiki.org/wiki/Algebraic_Dual_Separates_Points_of_Vector_Space | [
"Algebraic Duals"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Algebraic Dual/Vector Space",
"Definition:Mappings Separating Points"
] | [
"Definition:Mappings Separating Points",
"Definition:Linear Functional",
"Linearly Independent Set is Contained in some Basis",
"Definition:Basis of Vector Space",
"Linear Transformation Defined from Basis",
"Definition:Mappings Separating Points",
"Category:Algebraic Duals"
] |
proofwiki-22924 | Linear Transformation Defined from Basis | Let $K$ be a field.
Let $X$ and $Y$ be vector spaces over $K$.
Let $\BB = \set {e_\alpha : \alpha \in I}$ be a basis for $X$.
For each $\alpha \in I$, let $c_\alpha \in Y$.
Then there exists a linear transformation $T : X \to Y$ with $T e_\alpha = c_\alpha$ for each $\alpha \in I$.
This allows us to define a linear tr... | Define a mapping $T : X \to Y$ by:
:$\map T { {\mathbf 0}_X} = {\mathbf 0}_Y$
and:
:$\ds \map T {\sum_{\alpha \mathop \in F} \lambda_\alpha e_\alpha} = \sum_{\alpha \mathop \in F} \lambda_\alpha c_\alpha$ for every non-empty finite set $F \subseteq I$ and $\set {\lambda_\alpha : \alpha \in F} \subseteq K$.
Note that t... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ and $Y$ be [[Definition:Vector Space|vector spaces]] over $K$.
Let $\BB = \set {e_\alpha : \alpha \in I}$ be a [[Definition:Basis for Vector Space|basis]] for $X$.
For each $\alpha \in I$, let $c_\alpha \in Y$.
Then there exists a [[Definition:Li... | Define a [[Definition:Mapping|mapping]] $T : X \to Y$ by:
:$\map T { {\mathbf 0}_X} = {\mathbf 0}_Y$
and:
:$\ds \map T {\sum_{\alpha \mathop \in F} \lambda_\alpha e_\alpha} = \sum_{\alpha \mathop \in F} \lambda_\alpha c_\alpha$ for every [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set|finite set]] $F \su... | Linear Transformation Defined from Basis | https://proofwiki.org/wiki/Linear_Transformation_Defined_from_Basis | https://proofwiki.org/wiki/Linear_Transformation_Defined_from_Basis | [
"Linear Algebra"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Basis of Vector Space",
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Definition:Basis of Vector Space"
] | [
"Definition:Mapping",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Basis of Vector Space",
"Definition:Finite Set",
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Category:Linear Algebra"
] |
proofwiki-22925 | Absolute Polar Reverses Subset Relation | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $A \subseteq B \subseteq E$ be non-empty.
Let $A^\circ$ and $B^\circ$ denote the absolute polars of $A$ and $B$ in $\innerprod E F$.
Then $B^\circ \subseteq A^\circ$. | Suppose that $f \in B^\circ$.
Then $\cmod {\innerprod x f} \le 1$ for all $x \in B$.
Since $A \subseteq B$, we in particular have $\cmod {\innerprod x f} \le 1$ for all $x \in A$.
Hence $f \in A^\circ$.
We have shown that if $f \in B^\circ$, then $f \in A^\circ$, so $B^\circ \subseteq A^\circ$.
{{qed}} | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $A \subseteq B \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $A^\circ$ and $B^\circ$ denote the [[Definition:Absolute Polar|absolute polars]] of $A$ and $B$ in $\innerprod E F$.
Then $B^\circ... | Suppose that $f \in B^\circ$.
Then $\cmod {\innerprod x f} \le 1$ for all $x \in B$.
Since $A \subseteq B$, we in particular have $\cmod {\innerprod x f} \le 1$ for all $x \in A$.
Hence $f \in A^\circ$.
We have shown that if $f \in B^\circ$, then $f \in A^\circ$, so $B^\circ \subseteq A^\circ$.
{{qed}} | Absolute Polar Reverses Subset Relation | https://proofwiki.org/wiki/Absolute_Polar_Reverses_Subset_Relation | https://proofwiki.org/wiki/Absolute_Polar_Reverses_Subset_Relation | [
"Absolute Polars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Absolute Polar"
] | [] |
proofwiki-22926 | Absolute Polar of Dilation | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $A \subseteq E$ be non-empty.
Let $\lambda > 0$.
Let $A^\circ$ and $\paren {\lambda A}^\circ$ be the absolute polar of $A$ and $\lambda A$ respectively.
Then:
:$\paren {\lambda A}^\circ = \dfrac 1 \lambda A^\circ$ | Let $f \in \dfrac 1 \lambda A^\circ$.
This is the case {{iff}} $\lambda f \in A^\circ$.
This is equivalent to $\cmod {\innerprod x {\lambda f} } \le 1$ for each $x \in A$.
Since $\innerprod \cdot \cdot$ is bilinear, we have $\innerprod x {\lambda f} = \innerprod {\lambda x} f$.
Hence $\cmod {\innerprod x {\lambda f} }... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $A \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $\lambda > 0$.
Let $A^\circ$ and $\paren {\lambda A}^\circ$ be the [[Definition:Absolute Polar|absolute polar]] of $A$ and $\lambda A$ respect... | Let $f \in \dfrac 1 \lambda A^\circ$.
This is the case {{iff}} $\lambda f \in A^\circ$.
This is equivalent to $\cmod {\innerprod x {\lambda f} } \le 1$ for each $x \in A$.
Since $\innerprod \cdot \cdot$ is [[Definition:Bilinear Mapping|bilinear]], we have $\innerprod x {\lambda f} = \innerprod {\lambda x} f$.
Henc... | Absolute Polar of Dilation | https://proofwiki.org/wiki/Absolute_Polar_of_Dilation | https://proofwiki.org/wiki/Absolute_Polar_of_Dilation | [
"Absolute Polars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Absolute Polar"
] | [
"Definition:Bilinear Mapping"
] |
proofwiki-22927 | Intersection of Absolute Polars is Absolute Polar of Union | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $\FF$ be a set of non-empty subsets of $E$.
Then:
:$\ds \bigcap_{A \in \FF} A^\circ = \paren {\bigcup_{A \in \FF} A}^\circ$
where $\circ$ denotes absolute polar. | We have:
:$\ds f \in \paren {\bigcup_{A \in \FF} A}^\circ$
{{iff}}:
:$\cmod {\innerprod x f} \le 1$ for each $\ds x \in \bigcup_{A \in \FF} A$
This is equivalent to:
:$\cmod {\innerprod x f} \le 1$ for each $\ds x \in A$
for each $A \in \FF$.
This is equivalent to $f \in A^\circ$ for each $A \in \FF$.
This is equivalen... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $\FF$ be a [[Definition:Set|set]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subsets]] of $E$.
Then:
:$\ds \bigcap_{A \in \FF} A^\circ = \paren {\bigcup_{A \in \FF} A}^\circ$
where $\cir... | We have:
:$\ds f \in \paren {\bigcup_{A \in \FF} A}^\circ$
{{iff}}:
:$\cmod {\innerprod x f} \le 1$ for each $\ds x \in \bigcup_{A \in \FF} A$
This is equivalent to:
:$\cmod {\innerprod x f} \le 1$ for each $\ds x \in A$
for each $A \in \FF$.
This is equivalent to $f \in A^\circ$ for each $A \in \FF$.
This is equiva... | Intersection of Absolute Polars is Absolute Polar of Union | https://proofwiki.org/wiki/Intersection_of_Absolute_Polars_is_Absolute_Polar_of_Union | https://proofwiki.org/wiki/Intersection_of_Absolute_Polars_is_Absolute_Polar_of_Union | [
"Absolute Polars"
] | [
"Definition:Dual System",
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Absolute Polar"
] | [] |
proofwiki-22928 | Absolute Polar is Non-Empty | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $A \subseteq E$ be non-empty.
Let $A^\circ$ be the absolute polar of $A$.
Then $A^\circ \ne \O$. | Since $\innerprod \cdot \cdot$ is bilinear, we have $\innerprod x { {\mathbf 0}_F} = 0$ for each $x \in E$.
In particular, we have $\innerprod x { {\mathbf 0}_F} = 0$ for each $x \in A$.
Hence ${\mathbf 0}_F \in A^\circ$, and $A^\circ \ne \O$.
{{qed}} | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $A \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $A^\circ$ be the [[Definition:Absolute Polar|absolute polar]] of $A$.
Then $A^\circ \ne \O$. | Since $\innerprod \cdot \cdot$ is [[Definition:Bilinear Mapping|bilinear]], we have $\innerprod x { {\mathbf 0}_F} = 0$ for each $x \in E$.
In particular, we have $\innerprod x { {\mathbf 0}_F} = 0$ for each $x \in A$.
Hence ${\mathbf 0}_F \in A^\circ$, and $A^\circ \ne \O$.
{{qed}} | Absolute Polar is Non-Empty | https://proofwiki.org/wiki/Absolute_Polar_is_Non-Empty | https://proofwiki.org/wiki/Absolute_Polar_is_Non-Empty | [
"Absolute Polars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Absolute Polar"
] | [
"Definition:Bilinear Mapping"
] |
proofwiki-22929 | Absolute Polar is Convex | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $A \subseteq E$ be non-empty.
Let $A^\circ$ be the absolute polar of $A$.
Then $A^\circ$ is convex. | Let $f, g \in A^\circ$ and $t \in \closedint 0 1$.
Then, for all $x \in A$ we have:
:$\cmod {\innerprod x f} \le 1$
and:
:$\cmod {\innerprod x g} \le 1$
Then we have:
{{begin-eqn}}
{{eqn | l = \cmod {\innerprod x {t f + \paren {1 - t} g} }
| r = \cmod {t \innerprod x f + \paren {1 - t} \innerprod x g}
| c = $\inner... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $A \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $A^\circ$ be the [[Definition:Absolute Polar|absolute polar]] of $A$.
Then $A^\circ$ is [[Definition:Convex Set (Vector Space)|convex]]. | Let $f, g \in A^\circ$ and $t \in \closedint 0 1$.
Then, for all $x \in A$ we have:
:$\cmod {\innerprod x f} \le 1$
and:
:$\cmod {\innerprod x g} \le 1$
Then we have:
{{begin-eqn}}
{{eqn | l = \cmod {\innerprod x {t f + \paren {1 - t} g} }
| r = \cmod {t \innerprod x f + \paren {1 - t} \innerprod x g}
| c = $\inn... | Absolute Polar is Convex | https://proofwiki.org/wiki/Absolute_Polar_is_Convex | https://proofwiki.org/wiki/Absolute_Polar_is_Convex | [
"Convex Sets (Vector Spaces)",
"Absolute Polars",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Absolute Polar",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Bilinear Mapping",
"Definition:Convex Set (Vector Space)"
] |
proofwiki-22930 | Absolute Polar is Balanced | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $A \subseteq E$ be non-empty.
Let $A^\circ$ be the absolute polar of $A$.
Then $A^\circ$ is balanced. | Let $s \in \GF$ be such that $\cmod s \le 1$.
Let $f \in A^\circ$.
Then $\cmod {\innerprod x f} \le 1$ for each $x \in A$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {\innerprod x {s f} }
| r = \cmod {s \innerprod x f}
| c = $\innerprod \cdot \cdot$ is bilinear
}}
{{eqn | r = \cmod s \cmod {\innerprod x f}
}}
{{eqn | o... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $A \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $A^\circ$ be the [[Definition:Absolute Polar|absolute polar]] of $A$.
Then $A^\circ$ is [[Definition:Balanced Set|balanced]]. | Let $s \in \GF$ be such that $\cmod s \le 1$.
Let $f \in A^\circ$.
Then $\cmod {\innerprod x f} \le 1$ for each $x \in A$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {\innerprod x {s f} }
| r = \cmod {s \innerprod x f}
| c = $\innerprod \cdot \cdot$ is [[Definition:Bilinear Mapping|bilinear]]
}}
{{eqn | r = \cmod s ... | Absolute Polar is Balanced | https://proofwiki.org/wiki/Absolute_Polar_is_Balanced | https://proofwiki.org/wiki/Absolute_Polar_is_Balanced | [
"Absolute Polars",
"Balanced Sets"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Absolute Polar",
"Definition:Balanced Set"
] | [
"Definition:Bilinear Mapping"
] |
proofwiki-22931 | Characterization of Convergence of Net in Weak Topology Induced by Dual System | Let $K$ be a topological field.
Let $\innerprod X {X'}$ be a dual system over $K$.
Let $\map \sigma {X, X'}$ be the weak topology on $X$ induced by $\innerprod X {X'}$.
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 $\fam... | For each $x' \in X'$, define $f_{x'} : X \to \GF$ by:
:$\map {f_{x'} } x = \innerprod x {x'}$ for each $x \in X$.
Let $F = \set {f_{x'} : x' \in X'}$.
By the definition of the weak topology on $X$ induced by $\innerprod X {X'}$, $\map \sigma {X, X'}$ is the initial topology on $X$ induced by $F$.
By Characterization o... | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $\innerprod X {X'}$ be a [[Definition:Dual System|dual system]] over $K$.
Let $\map \sigma {X, X'}$ be the [[Definition:Weak Topology Induced by Dual System|weak topology on $X$ induced by $\innerprod X {X'}$]].
Let $\struct {\Lambda, \preceq}$ be... | For each $x' \in X'$, define $f_{x'} : X \to \GF$ by:
:$\map {f_{x'} } x = \innerprod x {x'}$ for each $x \in X$.
Let $F = \set {f_{x'} : x' \in X'}$.
By the definition of the [[Definition:Weak Topology Induced by Dual System|weak topology on $X$ induced by $\innerprod X {X'}$]], $\map \sigma {X, X'}$ is the [[Defin... | Characterization of Convergence of Net in Weak Topology Induced by Dual System | https://proofwiki.org/wiki/Characterization_of_Convergence_of_Net_in_Weak_Topology_Induced_by_Dual_System | https://proofwiki.org/wiki/Characterization_of_Convergence_of_Net_in_Weak_Topology_Induced_by_Dual_System | [
"Weak Topologies Induced by Dual Systems",
"Nets (Set Theory)"
] | [
"Definition:Topological Field",
"Definition:Dual System",
"Definition:Weak Topology Induced by Dual System",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net"
] | [
"Definition:Weak Topology Induced by Dual System",
"Definition:Initial Topology",
"Characterization of Convergence of Net in Initial Topology",
"Category:Weak Topologies Induced by Dual Systems",
"Category:Nets (Set Theory)"
] |
proofwiki-22932 | Open Set in Initial Topology on Vector Space Induced by Linear Functionals | Let $\GF \in \set {\R, \C}$.
Let $E$ be a vector space over $\GF$.
Let $E^\#$ be the algebraic dual of $E$.
Let $F \subseteq E^\#$ be a vector subspace.
Let $\tau$ be the initial topology on $E$ induced by $F$.
Let $U \subseteq E$.
Then $U$ is open in $\struct {X, \tau}$ {{iff}} for each $x \in X$ there exists $\epsil... | From Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex, $\tau$ is generated by the seminorms:
:$\set {p_f : f \in F}$
where we define $p_f : X \to \R_{\ge 0}$ by:
:$\map {p_f} x = \cmod {\map f x}$
for each $x \in X$.
Then, from Open Sets in Standard Topology of Locally Convex Space, $... | Let $\GF \in \set {\R, \C}$.
Let $E$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $E^\#$ be the [[Definition:Algebraic Dual|algebraic dual]] of $E$.
Let $F \subseteq E^\#$ be a [[Definition:Vector Subspace|vector subspace]].
Let $\tau$ be the [[Definition:Initial Topology|initial topology on $E$ in... | From [[Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex]], $\tau$ is [[Definition:Locally Convex Space/Standard Topology|generated]] by the [[Definition:Seminorm|seminorms]]:
:$\set {p_f : f \in F}$
where we define $p_f : X \to \R_{\ge 0}$ by:
:$\map {p_f} x = \cmod {\map f x}$
for ea... | Open Set in Initial Topology on Vector Space Induced by Linear Functionals | https://proofwiki.org/wiki/Open_Set_in_Initial_Topology_on_Vector_Space_Induced_by_Linear_Functionals | https://proofwiki.org/wiki/Open_Set_in_Initial_Topology_on_Vector_Space_Induced_by_Linear_Functionals | [
"Initial Topology"
] | [
"Definition:Vector Space",
"Definition:Algebraic Dual",
"Definition:Vector Subspace",
"Definition:Initial Topology",
"Definition:Open Set"
] | [
"Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Seminorm",
"Open Sets in Standard Topology of Locally Convex Space",
"Definition:Open Set",
"Definition:Linear Functional",
"Category:Initial Topology"
] |
proofwiki-22933 | Characterization of von Neumann-Boundedness in Weak Topology Induced by Dual System | Let $\GF \in \set {\R, \C}$.
Let $\innerprod X {X'}$ be a dual system over $\GF$.
Let $\map \sigma {X, X'}$ be the weak topology on $X$ induced by $X'$.
Let $E \subseteq X$.
Then $E$ is von Neumann-bounded in $\struct {X, \map \sigma {X, X'} }$ {{iff}} for each $x' \in X'$ we have:
:$\ds \sup_{x \mathop \in E} \cmod {\... | For each $x' \in X'$, define $f_{x'} : X \to \GF$ by:
:$\map {f_{x'} } x = \innerprod x {x'}$ for each $x \in X$.
Let $F = \set {f_{x'} : x' \in X'}$.
By the definition of the weak topology on $X$ induced by $X'$, $\map \sigma {X, X'}$ is the initial topology on $X$ generated by $F$.
By Characterization of von Neumann-... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod X {X'}$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $\map \sigma {X, X'}$ be the [[Definition:Weak Topology Induced by Dual System|weak topology on $X$ induced by $X'$]].
Let $E \subseteq X$.
Then $E$ is [[Definition:Von Neumann-Bounded Subset of Topolog... | For each $x' \in X'$, define $f_{x'} : X \to \GF$ by:
:$\map {f_{x'} } x = \innerprod x {x'}$ for each $x \in X$.
Let $F = \set {f_{x'} : x' \in X'}$.
By the definition of the [[Definition:Weak Topology Induced by Dual System|weak topology on $X$ induced by $X'$]], $\map \sigma {X, X'}$ is the [[Definition:Initial To... | Characterization of von Neumann-Boundedness in Weak Topology Induced by Dual System | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Boundedness_in_Weak_Topology_Induced_by_Dual_System | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Boundedness_in_Weak_Topology_Induced_by_Dual_System | [
"Von Neumann-Bounded Subsets of Topological Vector Spaces",
"Dual Systems"
] | [
"Definition:Dual System",
"Definition:Weak Topology Induced by Dual System",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Definition:Weak Topology Induced by Dual System",
"Definition:Initial Topology",
"Characterization of von Neumann-Boundedness in Initial Topology on Vector Space Induced by Linear Functionals",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space",
"Category:Von Neumann-Bounded Subsets of Top... |
proofwiki-22934 | Characterization of von Neumann-Boundedness in Weak-* Topology Induced by Dual System | Let $\GF \in \set {\R, \C}$.
Let $\innerprod X {X'}$ be a dual system over $\GF$.
Let $\map \sigma {X', X}$ be the weak topology on $X'$ induced by $X$.
Let $E' \subseteq X'$.
Then $E'$ is von Neumann-bounded in $\struct {X', \map \sigma {X', X} }$ {{iff}} for each $x \in X$ we have:
:$\ds \sup_{x' \mathop \in E'} \cmo... | For each $x \in X$, define $\pi_x : X' \to K$ by:
:$\map {\pi_x} {x'} = \innerprod x {x'}$ for each $x \in X$.
Let:
:$F = \set {\pi_x : x \in X}$
By the definition of the weak topology on $X'$ induced by $\innerprod X {X'}$, $\map \sigma {X', X}$ is the initial topology on $X$ generated by $F$.
By Characterization of v... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod X {X'}$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $\map \sigma {X', X}$ be the [[Definition:Weak Topology Induced by Dual System|weak topology on $X'$ induced by $X$]].
Let $E' \subseteq X'$.
Then $E'$ is [[Definition:Von Neumann-Bounded Subset of Topo... | For each $x \in X$, define $\pi_x : X' \to K$ by:
:$\map {\pi_x} {x'} = \innerprod x {x'}$ for each $x \in X$.
Let:
:$F = \set {\pi_x : x \in X}$
By the definition of the [[Definition:Weak Topology Induced by Dual System|weak topology on $X'$ induced by $\innerprod X {X'}$]], $\map \sigma {X', X}$ is the [[Definition... | Characterization of von Neumann-Boundedness in Weak-* Topology Induced by Dual System | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Boundedness_in_Weak-*_Topology_Induced_by_Dual_System | https://proofwiki.org/wiki/Characterization_of_von_Neumann-Boundedness_in_Weak-*_Topology_Induced_by_Dual_System | [
"Von Neumann-Bounded Subsets of Topological Vector Spaces",
"Dual Systems"
] | [
"Definition:Dual System",
"Definition:Weak Topology Induced by Dual System",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Definition:Weak Topology Induced by Dual System",
"Definition:Initial Topology",
"Characterization of von Neumann-Boundedness in Initial Topology on Vector Space Induced by Linear Functionals",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space",
"Category:Von Neumann-Bounded Subsets of Top... |
proofwiki-22935 | Transpose of Dual System is Dual System | Let $K$ be a field.
Let $\innerprod E F_{E \times F}$ be a dual system over $K$.
Define a mapping $\innerprod \cdot \cdot_{F \times E} : F \times E \to K$ by:
:$\innerprod f x_{F \times E} = \innerprod x f_{E \times F}$
Then $\innerprod F E_{F \times E}$ is a dual system over $K$. | By Transpose of Bilinear Mapping is Bilinear Mapping, $\innerprod \cdot \cdot_{F \times E}$ is a bilinear mapping.
Suppose that $f_0 \in F$ is such that:
:$\innerprod {f_0} x_{F \times E} = {\mathbf 0}_K$
for each $x \in E$.
Then we have:
:$\innerprod x {f_0}_{E \times F} = {\mathbf 0}_K$
Since $\innerprod \cdot \cdot_... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\innerprod E F_{E \times F}$ be a [[Definition:Dual System|dual system]] over $K$.
Define a [[Definition:Mapping|mapping]] $\innerprod \cdot \cdot_{F \times E} : F \times E \to K$ by:
:$\innerprod f x_{F \times E} = \innerprod x f_{E \times F}$
Then $... | By [[Transpose of Bilinear Mapping is Bilinear Mapping]], $\innerprod \cdot \cdot_{F \times E}$ is a [[Definition:Bilinear Mapping|bilinear mapping]].
Suppose that $f_0 \in F$ is such that:
:$\innerprod {f_0} x_{F \times E} = {\mathbf 0}_K$
for each $x \in E$.
Then we have:
:$\innerprod x {f_0}_{E \times F} = {\mathb... | Transpose of Dual System is Dual System | https://proofwiki.org/wiki/Transpose_of_Dual_System_is_Dual_System | https://proofwiki.org/wiki/Transpose_of_Dual_System_is_Dual_System | [
"Dual Systems"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Dual System",
"Definition:Mapping",
"Definition:Dual System"
] | [
"Transpose of Bilinear Mapping is Bilinear Mapping",
"Definition:Bilinear Mapping",
"Definition:Dual System",
"Definition:Dual System",
"Category:Dual Systems"
] |
proofwiki-22936 | Absolute Prepolar is Absolute Polar in Transpose | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F_{E \times F}$ be a dual system over $\GF$.
Let $\innerprod F E_{F \times E}$ be the transpose dual system.
Let $B \subseteq F$ be a non-empty set.
Let $B_\circ$ be the absolute prepolar of $B$ taken in $\innerprod E F_{E \times F}$.
Let $B^{\circ^T}$ be the absolute pola... | We have:
{{begin-eqn}}
{{eqn | l = B_\circ
| r = \set {x \in E : \cmod {\innerprod x f_{E \times F} } \le 1 \text { for all } f \in B}
}}
{{eqn | r = \set {x \in E : \cmod {\innerprod f x_{F \times E} } \le 1 \text { for all } f \in B}
}}
{{eqn | r = B^{\circ^T}
}}
{{end-eqn}}
{{qed}}
Category:Absolute Polars
Categor... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F_{E \times F}$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $\innerprod F E_{F \times E}$ be the [[Definition:Transpose Dual System|transpose dual system]].
Let $B \subseteq F$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $B_\circ$ be the [[Defi... | We have:
{{begin-eqn}}
{{eqn | l = B_\circ
| r = \set {x \in E : \cmod {\innerprod x f_{E \times F} } \le 1 \text { for all } f \in B}
}}
{{eqn | r = \set {x \in E : \cmod {\innerprod f x_{F \times E} } \le 1 \text { for all } f \in B}
}}
{{eqn | r = B^{\circ^T}
}}
{{end-eqn}}
{{qed}}
[[Category:Absolute Polars]]
[[... | Absolute Prepolar is Absolute Polar in Transpose | https://proofwiki.org/wiki/Absolute_Prepolar_is_Absolute_Polar_in_Transpose | https://proofwiki.org/wiki/Absolute_Prepolar_is_Absolute_Polar_in_Transpose | [
"Absolute Polars",
"Absolute Prepolars"
] | [
"Definition:Dual System",
"Definition:Transpose Dual System",
"Definition:Non-Empty Set",
"Definition:Absolute Prepolar",
"Definition:Absolute Polar"
] | [
"Category:Absolute Polars",
"Category:Absolute Prepolars"
] |
proofwiki-22937 | Weak Topology on Transpose Dual System is Weak-* Topology on Original System | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F_{E \times F}$ be a dual system over $\GF$.
Let $\innerprod F E_{F \times E}$ be the transpose of $\innerprod E F_{E \times F}$.
Let $\tau_1$ be the weak topology on $F$ induced by $\innerprod F E_{F \times E}$.
Let $\tau_2$ be the weak-$\ast$ topology on $F$ induced by ... | For each $x \in E$, define the mapping $g_x : F \to K$ by:
:$\map {g_x} f = \innerprod f x_{F \times E} = \innerprod x f_{E \times F}$ for each $\tuple {f, x} \in F \times E$.
By the definition of the weak topology on $F$, $\tau_1$ is the initial topology induced by $\set {g_x : x \in E}$.
For each $x \in E$, define a ... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F_{E \times F}$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $\innerprod F E_{F \times E}$ be the [[Definition:Transpose of Dual System|transpose]] of $\innerprod E F_{E \times F}$.
Let $\tau_1$ be the [[Definition:Weak Topology Induced by Dual System|wea... | For each $x \in E$, define the [[Definition:Mapping|mapping]] $g_x : F \to K$ by:
:$\map {g_x} f = \innerprod f x_{F \times E} = \innerprod x f_{E \times F}$ for each $\tuple {f, x} \in F \times E$.
By the definition of the [[Definition:Weak Topology Induced by Dual System|weak topology]] on $F$, $\tau_1$ is [[Definit... | Weak Topology on Transpose Dual System is Weak-* Topology on Original System | https://proofwiki.org/wiki/Weak_Topology_on_Transpose_Dual_System_is_Weak-*_Topology_on_Original_System | https://proofwiki.org/wiki/Weak_Topology_on_Transpose_Dual_System_is_Weak-*_Topology_on_Original_System | [
"Weak Topologies Induced by Dual Systems",
"Weak-* Topologies Induced by Dual Systems"
] | [
"Definition:Dual System",
"Definition:Transpose of Dual System",
"Definition:Weak Topology Induced by Dual System",
"Definition:Weak-* Topology Induced by Dual System",
"Definition:Topology"
] | [
"Definition:Mapping",
"Definition:Weak Topology Induced by Dual System",
"Definition:Initial Topology",
"Definition:Mapping",
"Definition:Weak-* Topology Induced by Dual System",
"Definition:Initial Topology",
"Category:Weak Topologies Induced by Dual Systems",
"Category:Weak-* Topologies Induced by D... |
proofwiki-22938 | Bipolar Set contains Original Set | Let $\GF \in \set {\R, \C}$.
Let $\innerprod X {X'}$ be a dual system over $\GF$.
Let $A \subseteq X$ be a non-empty set.
Let $\paren {A^\circ}_\circ$ be the bipolar of $A$.
Then:
:$A \subseteq \paren {A^\circ}_\circ$ | We have $x \in \paren {A^\circ}_\circ$ {{iff}}:
:$\cmod {\innerprod x f} \le 1$ for each $f \in A^\circ$.
We have that $f \in A^\circ$ {{iff}}:
:$\cmod {\innerprod x f} \le 1$ for each $x \in A$.
Hence for $x \in A$, we have:
:$\cmod {\innerprod x f} \le 1$ for each $f \in A^\circ$.
Hence $x \in \paren {A^\circ}_\circ... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod X {X'}$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $A \subseteq X$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $\paren {A^\circ}_\circ$ be the [[Definition:Bipolar Set|bipolar]] of $A$.
Then:
:$A \subseteq \paren {A^\circ}_\circ$ | We have $x \in \paren {A^\circ}_\circ$ {{iff}}:
:$\cmod {\innerprod x f} \le 1$ for each $f \in A^\circ$.
We have that $f \in A^\circ$ {{iff}}:
:$\cmod {\innerprod x f} \le 1$ for each $x \in A$.
Hence for $x \in A$, we have:
:$\cmod {\innerprod x f} \le 1$ for each $f \in A^\circ$.
Hence $x \in \paren {A^\circ}_\c... | Bipolar Set contains Original Set | https://proofwiki.org/wiki/Bipolar_Set_contains_Original_Set | https://proofwiki.org/wiki/Bipolar_Set_contains_Original_Set | [
"Bipolar Sets"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Bipolar Set"
] | [
"Category:Bipolar Sets"
] |
proofwiki-22939 | Alaoglu-Bourbaki Theorem | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $\map \sigma {E, F}$ be the weak topology on $E$ induced by $\innerprod E F$.
Let $V \subseteq E$ be a neighborhood of ${\mathbf 0}_X$ in $\map \sigma {E, F}$.
Let $V^\circ$ be the absolute polar of $V$.
Let $\map \sigma {F, E}$ be the w... | Let:
:$B = \set {z \in \C : \cmod z \le 1}$
By the Heine-Borel Theorem: Normed Vector Space, we have that $B$ is compact.
Hence by Tychonoff's Theorem, we have that:
:$\ds X = \prod_{x \mathop \in V} B$, equipped with the product topology $\tau_X$, is compact.
For each $f \in V^\circ$, define $\pi_f : E \to \GF$ by:
:... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $\map \sigma {E, F}$ be the [[Definition:Weak Topology Induced by Dual System|weak topology]] on $E$ induced by $\innerprod E F$.
Let $V \subseteq E$ be a [[Definition:Neighborhood (Topology)|neighborhood]] ... | Let:
:$B = \set {z \in \C : \cmod z \le 1}$
By the [[Heine-Borel Theorem/Normed Vector Space|Heine-Borel Theorem: Normed Vector Space]], we have that $B$ is [[Definition:Compact Topological Space|compact]].
Hence by [[Tychonoff's Theorem]], we have that:
:$\ds X = \prod_{x \mathop \in V} B$, equipped with the [[Defi... | Alaoglu-Bourbaki Theorem | https://proofwiki.org/wiki/Alaoglu-Bourbaki_Theorem | https://proofwiki.org/wiki/Alaoglu-Bourbaki_Theorem | [
"Dual Systems",
"Weak-* Topologies Induced by Dual Systems"
] | [
"Definition:Dual System",
"Definition:Weak Topology Induced by Dual System",
"Definition:Neighborhood (Topology)",
"Definition:Absolute Polar",
"Definition:Weak-* Topology Induced by Dual System",
"Definition:Compact Topological Space"
] | [
"Heine-Borel Theorem/Normed Vector Space",
"Definition:Compact Topological Space",
"Tychonoff's Theorem",
"Definition:Product Topology",
"Definition:Compact Topological Space",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Homeomorphism",
"Definition:Injection",
"Definition:Open Neighbor... |
proofwiki-22940 | Equivalence of Definitions of Boolean Lattice/Definition 2 implies Definition 1 | Let $\struct {S, \vee, \wedge, \neg}$ be a Boolean algebra.
Let $\struct {S, \vee, \wedge, \preceq}$ be an ordered structure.
Suppose that:
:$\forall a, b \in S: a \wedge b \preceq a \vee b$
Then, $\struct {S, \vee, \wedge, \preceq}$ is a complemented distributive lattice. | By axiom $\paren {\text {BA}_1 3}$, let $\bot, \top \in S$ denote the identities of $\vee$ and $\wedge$, respectively.
We will first show that $\struct {S, \vee, \wedge, \preceq}$ is a lattice.
By definition 2, we must prove that, for all $a, b \in S$:
:$a \vee b$ is the join of $a$ and $b$
:$a \wedge b$ is the meet of... | Let $\struct {S, \vee, \wedge, \neg}$ be a [[Definition:Boolean Algebra|Boolean algebra]].
Let $\struct {S, \vee, \wedge, \preceq}$ be an [[Definition:Ordered Structure|ordered structure]].
Suppose that:
:$\forall a, b \in S: a \wedge b \preceq a \vee b$
Then, $\struct {S, \vee, \wedge, \preceq}$ is a [[Definition:... | By [[Axiom:Boolean Algebra/Axioms/Formulation 1|axiom $\paren {\text {BA}_1 3}$]], let $\bot, \top \in S$ denote the [[Definition:Identity Element|identities]] of $\vee$ and $\wedge$, respectively.
We will first show that $\struct {S, \vee, \wedge, \preceq}$ is a [[Definition:Lattice (Order Theory)|lattice]].
By [[De... | Equivalence of Definitions of Boolean Lattice/Definition 2 implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Boolean_Lattice/Definition_2_implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Boolean_Lattice/Definition_2_implies_Definition_1 | [] | [
"Definition:Boolean Algebra",
"Definition:Ordered Structure",
"Definition:Complemented Lattice",
"Definition:Distributive Lattice"
] | [
"Axiom:Boolean Algebra/Axioms/Formulation 1",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Lattice (Order Theory)",
"Definition:Lattice (Order Theory)/Definition 2",
"Definition:Join (Order Theory)",
"Definition:Meet (Order Theory)",
"Identities of Boolean Algebra are also Ze... |
proofwiki-22941 | Boolean Lattice as Boolean Algebra | Let $\struct {S, \vee, \wedge, \preceq}$ be a Boolean lattice by definition $1$.
Then, there is a unique unary operation:
:$\neg : S \to S$
such that $\struct {S, \vee, \wedge, \neg}$ forms a Boolean algebra.
In particular, for each $a \in S$, we will have $\neg a$ be the complement of $a$. | Let $\neg a$ be defined as the unique complement of $a$ in $S$, if such an element exists.
By definition of a lattice, $a \vee b$ and $a \wedge b$ are defined in $S$ for all $a, b \in S$.
Moreover, by Complement in Boolean Lattice is Unique, $\neg a$ exists in $S$ for every $a \in S$.
So, axiom $\paren {\text {BA}_1 0}... | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Boolean Lattice/Definition 1|Boolean lattice by definition $1$]].
Then, there is a [[Definition:Unique|unique]] [[Definition:Unary Operation|unary operation]]:
:$\neg : S \to S$
such that $\struct {S, \vee, \wedge, \neg}$ forms a [[Definition:Boolean Algebra|B... | Let $\neg a$ be defined as the [[Definition:Unique|unique]] [[Definition:Complement (Lattice Theory)|complement]] of $a$ in $S$, if such an element exists.
By definition of a [[Definition:Lattice (Order Theory)/Definition 1|lattice]], $a \vee b$ and $a \wedge b$ are defined in $S$ for all $a, b \in S$.
Moreover, by ... | Boolean Lattice as Boolean Algebra | https://proofwiki.org/wiki/Boolean_Lattice_as_Boolean_Algebra | https://proofwiki.org/wiki/Boolean_Lattice_as_Boolean_Algebra | [
"Boolean Algebra is Equivalent to Boolean Lattice",
"Boolean Algebras",
"Boolean Lattices"
] | [
"Definition:Boolean Lattice/Definition 1",
"Definition:Unique",
"Definition:Operation/Unary Operation",
"Definition:Boolean Algebra",
"Definition:Complement (Lattice Theory)"
] | [
"Definition:Unique",
"Definition:Complement (Lattice Theory)",
"Definition:Lattice (Order Theory)/Definition 1",
"Complement in Distributive Lattice is Unique/Corollary",
"Axiom:Boolean Algebra/Axioms/Formulation 1",
"Join is Commutative",
"Meet is Commutative",
"Axiom:Boolean Algebra/Axioms/Formulati... |
proofwiki-22942 | Equivalence of Definitions of Boolean Lattice/Definition 1 implies Definition 3 | Let $\struct {S, \vee, \wedge, \preceq}$ be a complemented distributive lattice.
Then, there is a unary operation $\neg$ on $S$ such that:
:$\paren 1: \quad \forall a, b \in S: a \preceq \neg b \iff a \wedge b = \bot$
:$\paren 2: \quad \forall a \in S: \neg \neg a = a$ | Fix $a, b \in S$, and suppose that $a \preceq \neg b$.
Then, by Meet Precedes Operands:
:$a \wedge b \preceq a \preceq \neg b$
:$a \wedge b \preceq b$
so by definition of meet:
:$a \wedge b \preceq \neg b \wedge b$
Since $\neg b$ is the complement of $b$:
:$\neg b \wedge b = \bot$
hence:
:$a \wedge b \preceq \bot$
But ... | Let $\struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Complemented Lattice|complemented]] [[Definition:Distributive Lattice|distributive lattice]].
Then, there is a [[Definition:Unary Operation|unary operation]] $\neg$ on $S$ such that:
:$\paren 1: \quad \forall a, b \in S: a \preceq \neg b \iff a \wedge b = \bot... | Fix $a, b \in S$, and suppose that $a \preceq \neg b$.
Then, by [[Meet Precedes Operands]]:
:$a \wedge b \preceq a \preceq \neg b$
:$a \wedge b \preceq b$
so by definition of [[Definition:Meet (Order Theory)|meet]]:
:$a \wedge b \preceq \neg b \wedge b$
Since $\neg b$ is the [[Definition:Complement (Lattice Theory)|c... | Equivalence of Definitions of Boolean Lattice/Definition 1 implies Definition 3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Boolean_Lattice/Definition_1_implies_Definition_3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Boolean_Lattice/Definition_1_implies_Definition_3 | [] | [
"Definition:Complemented Lattice",
"Definition:Distributive Lattice",
"Definition:Operation/Unary Operation"
] | [
"Meet Precedes Operands",
"Definition:Meet (Order Theory)",
"Definition:Complement (Lattice Theory)",
"Definition:Bottom of Lattice",
"Definition:Smallest Element",
"Join Succeeds Operands",
"Complement in Boolean Algebra is Unique",
"Definition:Uniquely Complemented Lattice",
"Complement of Complem... |
proofwiki-22943 | Mass of Body with given Density | Let $\BB$ be a body occupying a region $V$ of space.
Let $\map \rho {\mathbf r}$ denote the (mass) density of $\BB$ at the point within $V$ whose position vector is $\mathbf r$.
Then the mass $m$ of $\BB$ is given by:
:$m = \ds \int_V \map \rho {\mathbf r} \rd V$ | To find the total mass of $\BB$, divide it into many finite regions $\set {\Delta \BB_i}$ each with a volume $\Delta V_i$.
Within each $\Delta \BB_i$, choose a point $\mathbf r_i$.
Let the density of $\Delta \BB_i$ be uniform and given by $\map \rho {\mathbf r_i}$.
Thus, the mass of $\Delta \BB_i$ is:
:$\Delta m_i = \m... | Let $\BB$ be a [[Definition:Body|body]] occupying a [[Definition:Region|region]] $V$ of [[Definition:Ordinary Space|space]].
Let $\map \rho {\mathbf r}$ denote the [[Definition:Mass Density|(mass) density]] of $\BB$ at the [[Definition:Point|point]] within $V$ whose [[Definition:Position Vector|position vector]] is $\... | To find the total [[Definition:Mass|mass]] of $\BB$, [[Definition:Subdivision of Interval|divide]] it into many [[Definition:Finite|finite]] [[Definition:Region|regions]] $\set {\Delta \BB_i}$ each with a [[Definition:Volume|volume]] $\Delta V_i$.
Within each $\Delta \BB_i$, choose a [[Definition:Point|point]] $\math... | Mass of Body with given Density | https://proofwiki.org/wiki/Mass_of_Body_with_given_Density | https://proofwiki.org/wiki/Mass_of_Body_with_given_Density | [
"Mass Density",
"Mass"
] | [
"Definition:Body",
"Definition:Region",
"Definition:Ordinary Space",
"Definition:Mass Density",
"Definition:Point",
"Definition:Position Vector",
"Definition:Mass"
] | [
"Definition:Mass",
"Definition:Subdivision of Interval",
"Definition:Finite",
"Definition:Region",
"Definition:Volume",
"Definition:Point",
"Definition:Mass Density",
"Definition:Uniform",
"Definition:Mass",
"Definition:Mass",
"Definition:Summation",
"Definition:Mass",
"Definition:Mass",
"... |
proofwiki-22944 | Expression for Balanced Hull | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $A \subseteq X$ be non-empty..
Then the balanced hull $\map {\operatorname {bal} } A$ is well-defined and:
:$\ds \map {\operatorname {bal} } A = \bigcup_{\lambda \in \GF : \cmod \lambda \le 1} \lambda A$ | We first show that:
:$\ds B = \bigcup_{\lambda \in \GF : \cmod \lambda \le 1} \lambda A$
is a balanced set containing $A$.
Let $x \in B$ and $\mu \in \GF$ be such that $\cmod \mu \le 1$.
Since $x \in B$, there exists $\lambda \in \GF$ with $\cmod \lambda \le 1$ and $y \in A$ such that $x = \lambda y$.
Then $\mu x = \l... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $A \subseteq X$ be [[Definition:Non-Empty Set|non-empty]]..
Then the [[Definition:Balanced Hull|balanced hull]] $\map {\operatorname {bal} } A$ is well-defined and:
:$\ds \map {\operatorname {bal} } A = \bigcup_{\lamb... | We first show that:
:$\ds B = \bigcup_{\lambda \in \GF : \cmod \lambda \le 1} \lambda A$
is a [[Definition:Balanced Set|balanced set]] containing $A$.
Let $x \in B$ and $\mu \in \GF$ be such that $\cmod \mu \le 1$.
Since $x \in B$, there exists $\lambda \in \GF$ with $\cmod \lambda \le 1$ and $y \in A$ such that $x ... | Expression for Balanced Hull | https://proofwiki.org/wiki/Expression_for_Balanced_Hull | https://proofwiki.org/wiki/Expression_for_Balanced_Hull | [
"Balanced Sets"
] | [
"Definition:Vector Space",
"Definition:Non-Empty Set",
"Definition:Balanced Hull"
] | [
"Definition:Balanced Set",
"Definition:Balanced Set",
"Definition:Balanced Set",
"Definition:Balanced Set",
"Definition:Balanced Set",
"Definition:Smallest Element",
"Definition:Balanced Set",
"Definition:Balanced Hull",
"Category:Balanced Sets"
] |
proofwiki-22945 | Balanced Hull is Intersection of Balanced Sets containing Set | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $A \subseteq X$ be a non-empty set.
Let $\map {\operatorname {bal} } A$ be the balanced hull of $A$.
Then:
:$\ds \map {\operatorname {bal} } A = \bigcap \set {B \supseteq A: B \text { is balanced} }$ | Note that:
:$\map {\operatorname {bal} } A \in \set {B \supseteq A: B \text { is balanced} }$
and hence:
:$\set {B \supseteq A: B \text { is balanced} } \ne \O$
Hence:
:$\ds \bigcap \set {B \supseteq A: B \text { is balanced} }$ is well-defined.
Further, from Intersection of Balanced Sets in Vector Space is Balanced, w... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $A \subseteq X$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $\map {\operatorname {bal} } A$ be the [[Definition:Balanced Hull|balanced hull]] of $A$.
Then:
:$\ds \map {\operatorname {bal} } A = \bigcap \set ... | Note that:
:$\map {\operatorname {bal} } A \in \set {B \supseteq A: B \text { is balanced} }$
and hence:
:$\set {B \supseteq A: B \text { is balanced} } \ne \O$
Hence:
:$\ds \bigcap \set {B \supseteq A: B \text { is balanced} }$ is well-defined.
Further, from [[Intersection of Balanced Sets in Vector Space is Balance... | Balanced Hull is Intersection of Balanced Sets containing Set | https://proofwiki.org/wiki/Balanced_Hull_is_Intersection_of_Balanced_Sets_containing_Set | https://proofwiki.org/wiki/Balanced_Hull_is_Intersection_of_Balanced_Sets_containing_Set | [
"Balanced Sets"
] | [
"Definition:Vector Space",
"Definition:Non-Empty Set",
"Definition:Balanced Hull"
] | [
"Intersection of Balanced Sets in Vector Space is Balanced",
"Definition:Balanced Set",
"Definition:Smallest Element",
"Definition:Balanced Set",
"Category:Balanced Sets"
] |
proofwiki-22946 | Convex Hull of Balanced Set is Balanced | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $B \subseteq X$ be a balanced set.
Let $\map {\operatorname {conv} } B$ be the convex hull of $B$.
Then $\map {\operatorname {conv} } B$ is balanced. | Let $x \in \map {\operatorname {conv} } B$ and $\lambda \in \GF$ have $\cmod \lambda \le 1$.
Then there exists $x_1, \ldots, x_n \in B$ and $t_1, \ldots, t_n \in \closedint 0 1$ such that:
:$\ds x = \sum_{j \mathop = 1}^n t_j x_j$
Then:
:$\ds \lambda x = \sum_{j \mathop = 1}^n t_j \paren {\lambda x_j}$
Since $B$ is ba... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $B \subseteq X$ be a [[Definition:Balanced Set|balanced set]].
Let $\map {\operatorname {conv} } B$ be the [[Definition:Convex Hull|convex hull]] of $B$.
Then $\map {\operatorname {conv} } B$ is [[Definition:Balanced... | Let $x \in \map {\operatorname {conv} } B$ and $\lambda \in \GF$ have $\cmod \lambda \le 1$.
Then there exists $x_1, \ldots, x_n \in B$ and $t_1, \ldots, t_n \in \closedint 0 1$ such that:
:$\ds x = \sum_{j \mathop = 1}^n t_j x_j$
Then:
:$\ds \lambda x = \sum_{j \mathop = 1}^n t_j \paren {\lambda x_j}$
Since $B$ is... | Convex Hull of Balanced Set is Balanced | https://proofwiki.org/wiki/Convex_Hull_of_Balanced_Set_is_Balanced | https://proofwiki.org/wiki/Convex_Hull_of_Balanced_Set_is_Balanced | [
"Convex Sets (Vector Spaces)",
"Balanced Sets"
] | [
"Definition:Vector Space",
"Definition:Balanced Set",
"Definition:Convex Hull",
"Definition:Balanced Set"
] | [
"Definition:Balanced Set",
"Definition:Balanced Set",
"Category:Convex Sets (Vector Spaces)",
"Category:Balanced Sets"
] |
proofwiki-22947 | Convex Balanced Hull is Intersection of all Convex Balanced Sets containing Set | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $A \subseteq X$ be a non-empty set.
Let $\map {\operatorname {cobal} } A$ be the convex balanced hull of $A$.
We then have:
:$\ds \map {\operatorname {cobal} } A = \bigcap \set {C \supseteq A : C \text { is convex and balanced} }$ | Let:
:$\ds B = \bigcap \set {C \supseteq A : C \text { is convex and balanced} }$
From Convex Hull of Balanced Set is Balanced, the convex hull of the balanced hull of $A$, $\map {\operatorname {conv} } {\map {\operatorname {bal} } A}$ is balanced.
Hence:
:$\set {C \supseteq A : C \text { is convex and balanced} } \ne ... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $A \subseteq X$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $\map {\operatorname {cobal} } A$ be the [[Definition:Convex Balanced Hull|convex balanced hull]] of $A$.
We then have:
:$\ds \map {\operatorname {... | Let:
:$\ds B = \bigcap \set {C \supseteq A : C \text { is convex and balanced} }$
From [[Convex Hull of Balanced Set is Balanced]], the [[Definition:Convex Hull|convex hull]] of the [[Definition:Balanced Hull|balanced hull]] of $A$, $\map {\operatorname {conv} } {\map {\operatorname {bal} } A}$ is [[Definition:Balance... | Convex Balanced Hull is Intersection of all Convex Balanced Sets containing Set | https://proofwiki.org/wiki/Convex_Balanced_Hull_is_Intersection_of_all_Convex_Balanced_Sets_containing_Set | https://proofwiki.org/wiki/Convex_Balanced_Hull_is_Intersection_of_all_Convex_Balanced_Sets_containing_Set | [
"Convex Balanced Hulls"
] | [
"Definition:Vector Space",
"Definition:Non-Empty Set",
"Definition:Convex Balanced Hull"
] | [
"Convex Hull of Balanced Set is Balanced",
"Definition:Convex Hull",
"Definition:Balanced Hull",
"Definition:Balanced Set",
"Intersection of Convex Sets is Convex Set (Vector Spaces)",
"Definition:Convex Set (Vector Space)",
"Intersection of Balanced Sets in Vector Space is Balanced",
"Definition:Bala... |
proofwiki-22948 | Convex Balanced Hull of Finite Union of Convex Balanced Sets | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $K_1, \ldots, K_n \subseteq X$ be convex balanced sets.
Let $\operatorname {convbal}$ denote convex balanced hull.
Then:
:$\ds \map {\operatorname {convbal} } {\bigcup_{j \mathop = 1}^n K_j} = \set {\sum_{j \mathop = 1}^n \lambda_j x_j : \lambda_1, ... | Let:
:$\ds C = \set {\sum_{j \mathop = 1}^n \lambda_j x_j : \lambda_1, \ldots, \lambda_n \in \GF, \, x_j \in K_j \text { for all } 1 \le k \le n \text { and } \sum_{j \mathop = 1}^n \cmod {\lambda_j} \le 1}$
Let $x, y \in C$ and $t \in \closedint 0 1$.
Then there exists $x_j \in K_j$ ($1 \le j \le n$), $y_j \in K_j$ ($... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $K_1, \ldots, K_n \subseteq X$ be [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Balanced Set|balanced sets]].
Let $\operatorname {convbal}$ denote [[Definition:Convex Balanced Hull|convex balanced hull]]... | Let:
:$\ds C = \set {\sum_{j \mathop = 1}^n \lambda_j x_j : \lambda_1, \ldots, \lambda_n \in \GF, \, x_j \in K_j \text { for all } 1 \le k \le n \text { and } \sum_{j \mathop = 1}^n \cmod {\lambda_j} \le 1}$
Let $x, y \in C$ and $t \in \closedint 0 1$.
Then there exists $x_j \in K_j$ ($1 \le j \le n$), $y_j \in K_j$ ... | Convex Balanced Hull of Finite Union of Convex Balanced Sets | https://proofwiki.org/wiki/Convex_Balanced_Hull_of_Finite_Union_of_Convex_Balanced_Sets | https://proofwiki.org/wiki/Convex_Balanced_Hull_of_Finite_Union_of_Convex_Balanced_Sets | [
"Convex Balanced Hulls",
"Convex Sets (Vector Spaces)",
"Balanced Sets"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Balanced Set",
"Definition:Convex Balanced Hull"
] | [
"Definition:Balanced Set",
"Definition:Balanced Set",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Balanced Set",
"Category:Convex Balanced Hulls",
"Category:Convex Sets (Vector Spaces)",
"Category:Balanced Sets"
] |
proofwiki-22949 | Absolute Prepolar is Non-Empty | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $B \subseteq F$ be non-empty.
Let $B_\circ$ be the absolute prepolar of $B$.
Then $B_\circ \ne \O$. | Since $\innerprod \cdot \cdot$ is bilinear, we have $\innerprod { {\mathbf 0}_E} f = 0$ for each $f \in F$.
In particular, we have $\innerprod { {\mathbf 0}_E} f = 0$ for each $f \in B$.
Hence ${\mathbf 0}_E \in B_\circ$, hence $B_\circ \ne \O$.
{{qed}}
Category:Absolute Prepolars
cpzv07ghkmaalp1dz3n48s0870kn6d4 | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $B \subseteq F$ be [[Definition:Non-Empty Set|non-empty]].
Let $B_\circ$ be the [[Definition:Absolute Prepolar|absolute prepolar]] of $B$.
Then $B_\circ \ne \O$. | Since $\innerprod \cdot \cdot$ is [[Definition:Bilinear Mapping|bilinear]], we have $\innerprod { {\mathbf 0}_E} f = 0$ for each $f \in F$.
In particular, we have $\innerprod { {\mathbf 0}_E} f = 0$ for each $f \in B$.
Hence ${\mathbf 0}_E \in B_\circ$, hence $B_\circ \ne \O$.
{{qed}}
[[Category:Absolute Prepolars]]... | Absolute Prepolar is Non-Empty | https://proofwiki.org/wiki/Absolute_Prepolar_is_Non-Empty | https://proofwiki.org/wiki/Absolute_Prepolar_is_Non-Empty | [
"Absolute Prepolars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Absolute Prepolar"
] | [
"Definition:Bilinear Mapping",
"Category:Absolute Prepolars"
] |
proofwiki-22950 | Absolute Prepolar is Convex Set | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $B \subseteq F$ be non-empty.
Let $B_\circ$ be the absolute prepolar of $B$.
Then $B_\circ$ is convex. | Let $x, y \in B_\circ$ and $t \in \closedint 0 1$.
Then:
:$\cmod {\innerprod x f} \le 1$
and:
:$\cmod {\innerprod y f} \le 1$
for all $f \in B$.
We now have:
{{begin-eqn}}
{{eqn | l = \cmod {\innerprod {t x + \paren {1 - t} y} f}
| r = \cmod {t \innerprod x f + \paren {1 - t} \innerprod y f}
| c = $\innerprod \cdot... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $B \subseteq F$ be [[Definition:Non-Empty Set|non-empty]].
Let $B_\circ$ be the [[Definition:Absolute Prepolar|absolute prepolar]] of $B$.
Then $B_\circ$ is [[Definition:Convex Set (Vector Space)|convex]]... | Let $x, y \in B_\circ$ and $t \in \closedint 0 1$.
Then:
:$\cmod {\innerprod x f} \le 1$
and:
:$\cmod {\innerprod y f} \le 1$
for all $f \in B$.
We now have:
{{begin-eqn}}
{{eqn | l = \cmod {\innerprod {t x + \paren {1 - t} y} f}
| r = \cmod {t \innerprod x f + \paren {1 - t} \innerprod y f}
| c = $\innerprod \cd... | Absolute Prepolar is Convex Set | https://proofwiki.org/wiki/Absolute_Prepolar_is_Convex_Set | https://proofwiki.org/wiki/Absolute_Prepolar_is_Convex_Set | [
"Absolute Prepolars",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Absolute Prepolar",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Bilinear Mapping",
"Triangle Inequality/Complex Numbers",
"Category:Absolute Prepolars",
"Category:Convex Sets (Vector Spaces)"
] |
proofwiki-22951 | Absolute Prepolar is Balanced | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $B \subseteq F$ be non-empty.
Let $B_\circ$ be the absolute prepolar of $B$.
Then $B_\circ$ is balanced. | Let $x \in B_\circ$ and $\lambda \in \GF$ have $\cmod \lambda \le 1$.
Then, for each $f \in B$, we have:
{{begin-eqn}}
{{eqn | l = \cmod {\innerprod {\lambda x} f}
| r = \cmod \lambda \cmod {\innerprod x f}
}}
{{eqn | o = \le
| r = \cmod {\innerprod x f}
}}
{{eqn | o = \le
| r = 1
}}
{{end-eqn}}
Hence $\lambda x ... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $B \subseteq F$ be [[Definition:Non-Empty Set|non-empty]].
Let $B_\circ$ be the [[Definition:Absolute Prepolar|absolute prepolar]] of $B$.
Then $B_\circ$ is [[Definition:Balanced Set|balanced]]. | Let $x \in B_\circ$ and $\lambda \in \GF$ have $\cmod \lambda \le 1$.
Then, for each $f \in B$, we have:
{{begin-eqn}}
{{eqn | l = \cmod {\innerprod {\lambda x} f}
| r = \cmod \lambda \cmod {\innerprod x f}
}}
{{eqn | o = \le
| r = \cmod {\innerprod x f}
}}
{{eqn | o = \le
| r = 1
}}
{{end-eqn}}
Hence $\lambda ... | Absolute Prepolar is Balanced | https://proofwiki.org/wiki/Absolute_Prepolar_is_Balanced | https://proofwiki.org/wiki/Absolute_Prepolar_is_Balanced | [
"Absolute Prepolars",
"Balanced Sets"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Absolute Prepolar",
"Definition:Balanced Set"
] | [
"Category:Absolute Prepolars",
"Category:Balanced Sets"
] |
proofwiki-22952 | Absolute Prepolar is Closed in Weak Topology | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $\map \sigma {E, F}$ be the weak-$\ast$ topology on $F$ induced by $\innerprod E F$.
Let $B \subseteq E$ be non-empty.
Let $B_\circ$ be the absolute prepolar of $B$.
Then $B_\circ$ is $\map \sigma {E, F}$-closed. | We use Characterization of Closedness in terms of Nets.
That is, we show that every convergent net valued in $B_\circ$ has its limit in $B_\circ$.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a convergent net valued in $B_\circ$ with limit $x$.
We show ... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $\map \sigma {E, F}$ be the [[Definition:Weak Topology Induced by Dual System|weak-$\ast$ topology on $F$ induced by $\innerprod E F$]].
Let $B \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $... | We use [[Characterization of Closedness in terms of Nets]].
That is, we show that every [[Definition:Convergent Net|convergent net]] valued in $B_\circ$ has its [[Definition:Limit of Net|limit]] in $B_\circ$.
Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {x_\lambda}_{\... | Absolute Prepolar is Closed in Weak Topology | https://proofwiki.org/wiki/Absolute_Prepolar_is_Closed_in_Weak_Topology | https://proofwiki.org/wiki/Absolute_Prepolar_is_Closed_in_Weak_Topology | [
"Weak-* Topologies Induced by Dual Systems",
"Weak Topologies Induced by Dual Systems",
"Absolute Prepolars",
"Weak Topologies Induced by Dual Systems"
] | [
"Definition:Dual System",
"Definition:Weak Topology Induced by Dual System",
"Definition:Non-Empty Set",
"Definition:Absolute Prepolar",
"Definition:Closed Set"
] | [
"Characterization of Closedness in terms of Nets",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Definition:Directed Preordering",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Characterization of Convergence of Net in Weak Topology Induced by Dual System",
"Category:Absolute Pre... |
proofwiki-22953 | Convex Real Function has Minimum at Point iff Zero is Subgradient at Point | Let $\innerprod X {X'}$ be a dual system over $\R$.
Let $C \subseteq X$ be a convex set.
Let $f : C \to \R$ be a convex function.
Let $x \in C$.
Then $f$ has a minimum at $x$ {{iff}} ${\mathbf 0}_{X'}$ is a subgradient at $x$ with respect to $\innerprod X {X'}$. | We have that $f$ is a minimum at $x$ {{iff}}:
:$\map f y \ge \map f x$ for all $y \in C$.
That is:
:$\map f y - \map f x \ge 0$
We have:
:$\innerprod {y - x} { {\mathbf 0}_{X'} } = 0$
Hence:
:$\map f y - \map f x \ge \innerprod {y - x} { {\mathbf 0}_{X'} }$ for all $y \in C$.
Hence ${\mathbf 0}_{X'}$ is a subgradient a... | Let $\innerprod X {X'}$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $C \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex set]].
Let $f : C \to \R$ be a [[Definition:Convex Real Function|convex function]].
Let $x \in C$.
Then $f$ has a [[Definition:Minimum Point|minimum]] at $x$ {{iff}}... | We have that $f$ is a [[Definition:Minimum Point|minimum]] at $x$ {{iff}}:
:$\map f y \ge \map f x$ for all $y \in C$.
That is:
:$\map f y - \map f x \ge 0$
We have:
:$\innerprod {y - x} { {\mathbf 0}_{X'} } = 0$
Hence:
:$\map f y - \map f x \ge \innerprod {y - x} { {\mathbf 0}_{X'} }$ for all $y \in C$.
Hence ${\m... | Convex Real Function has Minimum at Point iff Zero is Subgradient at Point | https://proofwiki.org/wiki/Convex_Real_Function_has_Minimum_at_Point_iff_Zero_is_Subgradient_at_Point | https://proofwiki.org/wiki/Convex_Real_Function_has_Minimum_at_Point_iff_Zero_is_Subgradient_at_Point | [
"Subgradients"
] | [
"Definition:Dual System",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Real Function",
"Definition:Minimum Point",
"Definition:Subgradient"
] | [
"Definition:Minimum Point",
"Definition:Subgradient"
] |
proofwiki-22954 | Bipolar Theorem/Absolute Polar | Let $\GF \in \set {\R, \C}$.
Let $\innerprod X {X'}$ be a dual system over $\GF$.
Let $A \subseteq X$ be a non-empty set.
Let $\paren {A^\circ}_\circ$ be the bipolar of $A$.
Let $\map \sigma {X, X'}$ be the weak topology on $X$ induced by $\innerprod X {X'}$.
Let $\map {\operatorname {convbal} } A$ be the convex bala... | From Bipolar Set is Convex, $\paren {A^\circ}_\circ$ is convex.
From Bipolar Set is Balanced, $\paren {A^\circ}_\circ$ is balanced.
From Bipolar Set is Closed in Weak Topology, $\paren {A^\circ}_\circ$ is $\map \sigma {E, F}$-closed.
From Bipolar Set contains Original Set, we have $A \subseteq \paren {A^\circ}_\circ$.
... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod X {X'}$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $A \subseteq X$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $\paren {A^\circ}_\circ$ be the [[Definition:Bipolar Set|bipolar]] of $A$.
Let $\map \sigma {X, X'}$ be the [[Definition:Weak Topolog... | From [[Bipolar Set is Convex]], $\paren {A^\circ}_\circ$ is [[Definition:Convex Set (Vector Space)|convex]].
From [[Bipolar Set is Balanced]], $\paren {A^\circ}_\circ$ is [[Definition:Balanced Set|balanced]].
From [[Bipolar Set is Closed in Weak Topology]], $\paren {A^\circ}_\circ$ is [[Definition:Closed Set|$\map \s... | Bipolar Theorem/Absolute Polar | https://proofwiki.org/wiki/Bipolar_Theorem/Absolute_Polar | https://proofwiki.org/wiki/Bipolar_Theorem/Absolute_Polar | [
"Bipolar Theorem",
"Absolute Polars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Bipolar Set",
"Definition:Weak Topology Induced by Dual System",
"Definition:Convex Balanced Hull",
"Definition:Closure (Topology)"
] | [
"Bipolar Set is Convex",
"Definition:Convex Set (Vector Space)",
"Bipolar Set is Balanced",
"Definition:Balanced Set",
"Bipolar Set is Closed in Weak Topology",
"Definition:Closed Set",
"Bipolar Set contains Original Set",
"Closure of Convex Balanced Hull as Intersection of Closed Convex Balanced Sets... |
proofwiki-22955 | Bipolar Set is Convex | vLet $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $A \subseteq E$ be non-empty.
Let $\paren {A^\circ}_\circ$ be the bipolar set of $A$.
Then $\paren {A^\circ}_\circ$ is convex. | Let $A^\circ$ be the absolute polar of $A$.
From Absolute Polar is Non-Empty, $A^\circ$ is non-empty.
From Absolute Prepolar is Convex Set, $\paren {A^\circ}_\circ$ is convex.
{{qed}}
Category:Bipolar Sets
5soc331wuibt5vnuqve15nrs970yy28 | vLet $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $A \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $\paren {A^\circ}_\circ$ be the [[Definition:Bipolar Set|bipolar set]] of $A$.
Then $\paren {A^\circ}_\circ$ is [[Definition:Convex Set (Vec... | Let $A^\circ$ be the [[Definition:Absolute Polar|absolute polar]] of $A$.
From [[Absolute Polar is Non-Empty]], $A^\circ$ is [[Definition:Non-Empty Set|non-empty]].
From [[Absolute Prepolar is Convex Set]], $\paren {A^\circ}_\circ$ is [[Definition:Convex Set (Vector Space)|convex]].
{{qed}}
[[Category:Bipolar Sets]]... | Bipolar Set is Convex | https://proofwiki.org/wiki/Bipolar_Set_is_Convex | https://proofwiki.org/wiki/Bipolar_Set_is_Convex | [
"Bipolar Sets"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Bipolar Set",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Absolute Polar",
"Absolute Polar is Non-Empty",
"Definition:Non-Empty Set",
"Absolute Prepolar is Convex Set",
"Definition:Convex Set (Vector Space)",
"Category:Bipolar Sets"
] |
proofwiki-22956 | Bipolar Set is Balanced | vLet $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $A \subseteq E$ be non-empty.
Let $\paren {A^\circ}_\circ$ be the bipolar set of $A$.
Then $\paren {A^\circ}_\circ$ is balanced. | Let $A^\circ$ be the absolute polar of $A$.
From Absolute Polar is Non-Empty, $A^\circ$ is non-empty.
From Absolute Prepolar is Balanced, $\paren {A^\circ}_\circ$ is balanced.
{{qed}}
Category:Bipolar Sets
Category:Balanced Sets
3n36esx3beyn7y6st7ce58giq3ll55o | vLet $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $A \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $\paren {A^\circ}_\circ$ be the [[Definition:Bipolar Set|bipolar set]] of $A$.
Then $\paren {A^\circ}_\circ$ is [[Definition:Balanced Set|ba... | Let $A^\circ$ be the [[Definition:Absolute Polar|absolute polar]] of $A$.
From [[Absolute Polar is Non-Empty]], $A^\circ$ is [[Definition:Non-Empty Set|non-empty]].
From [[Absolute Prepolar is Balanced]], $\paren {A^\circ}_\circ$ is [[Definition:Balanced Set|balanced]].
{{qed}}
[[Category:Bipolar Sets]]
[[Category:B... | Bipolar Set is Balanced | https://proofwiki.org/wiki/Bipolar_Set_is_Balanced | https://proofwiki.org/wiki/Bipolar_Set_is_Balanced | [
"Bipolar Sets",
"Balanced Sets"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:Bipolar Set",
"Definition:Balanced Set"
] | [
"Definition:Absolute Polar",
"Absolute Polar is Non-Empty",
"Definition:Non-Empty Set",
"Absolute Prepolar is Balanced",
"Definition:Balanced Set",
"Category:Bipolar Sets",
"Category:Balanced Sets"
] |
proofwiki-22957 | One-Sided Polar of Dilation | Let $\innerprod E F$ be a dual system over $\R$.
Let $A \subseteq E$ be non-empty.
Let $\lambda > 0$.
Let $A^\odot$ and $\paren {\lambda A}^\odot$ be the one-sided polar of $A$ and $\lambda A$ respectively.
Then:
:$\paren {\lambda A}^\odot = \dfrac 1 \lambda A^\odot$ | Let $f \in \dfrac 1 \lambda A^\odot$.
This is the case {{iff}} $\lambda f \in A^\odot$.
This is equivalent to $\innerprod x {\lambda f} \le 1$ for each $x \in A$.
Since $\innerprod \cdot \cdot$ is bilinear, we have $\innerprod x {\lambda f} = \innerprod {\lambda x} f$.
Hence $\innerprod x {\lambda f} \le 1$ for each $... | Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $A \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $\lambda > 0$.
Let $A^\odot$ and $\paren {\lambda A}^\odot$ be the [[Definition:One-Sided Polar|one-sided polar]] of $A$ and $\lambda A$ respectively.
Then:
:$\paren {\lam... | Let $f \in \dfrac 1 \lambda A^\odot$.
This is the case {{iff}} $\lambda f \in A^\odot$.
This is equivalent to $\innerprod x {\lambda f} \le 1$ for each $x \in A$.
Since $\innerprod \cdot \cdot$ is [[Definition:Bilinear Mapping|bilinear]], we have $\innerprod x {\lambda f} = \innerprod {\lambda x} f$.
Hence $\inner... | One-Sided Polar of Dilation | https://proofwiki.org/wiki/One-Sided_Polar_of_Dilation | https://proofwiki.org/wiki/One-Sided_Polar_of_Dilation | [
"One-Sided Polars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Polar"
] | [
"Definition:Bilinear Mapping"
] |
proofwiki-22958 | Intersection of One-Sided Polars is One-Sided Polar of Union | Let $\innerprod E F$ be a dual system over $\R$.
Let $\FF$ be a set of non-empty subsets of $E$.
Then:
:$\ds \bigcap_{A \in \FF} A^\odot = \paren {\bigcup_{A \in \FF} A}^\odot$
where $\odot$ denotes one-sided polar. | We have:
:$\ds f \in \paren {\bigcup_{A \in \FF} A}^\odot$
{{iff}}:
:$\innerprod x f \le 1$ for each $\ds x \in \bigcup_{A \in \FF} A$
This is equivalent to:
:$\innerprod x f \le 1$ for each $\ds x \in A$
for each $A \in \FF$.
This is equivalent to $f \in A^\odot$ for each $A \in \FF$.
This is equivalent to $\ds f \in ... | Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $\FF$ be a [[Definition:Set|set]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subsets]] of $E$.
Then:
:$\ds \bigcap_{A \in \FF} A^\odot = \paren {\bigcup_{A \in \FF} A}^\odot$
where $\odot$ denotes [[Definition:One-Sid... | We have:
:$\ds f \in \paren {\bigcup_{A \in \FF} A}^\odot$
{{iff}}:
:$\innerprod x f \le 1$ for each $\ds x \in \bigcup_{A \in \FF} A$
This is equivalent to:
:$\innerprod x f \le 1$ for each $\ds x \in A$
for each $A \in \FF$.
This is equivalent to $f \in A^\odot$ for each $A \in \FF$.
This is equivalent to $\ds f \... | Intersection of One-Sided Polars is One-Sided Polar of Union | https://proofwiki.org/wiki/Intersection_of_One-Sided_Polars_is_One-Sided_Polar_of_Union | https://proofwiki.org/wiki/Intersection_of_One-Sided_Polars_is_One-Sided_Polar_of_Union | [
"One-Sided Polars"
] | [
"Definition:Dual System",
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:One-Sided Polar"
] | [] |
proofwiki-22959 | One-Sided Polar is Non-Empty | Let $\innerprod E F$ be a dual system over $\R$.
Let $A \subseteq E$ be non-empty.
Let $A^\circ$ be the one-sided polar of $A$ in $\innerprod E F$.
Then $A^\circ \ne \O$. | Note that we have:
:$\innerprod x { {\mathbf 0}_F} = 0$ for all $x \in E$.
In particular:
:$\innerprod x { {\mathbf 0}_F} \le 1$ for all $x \in A$.
Hence we have ${\mathbf 0}_F \in A^\circ$.
Hence $A^\circ \ne \O$.
{{qed}}
Category:One-Sided Polars
banrxb2c7tm0lg4a7kgrrg0vysbpysc | Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $A \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $A^\circ$ be the [[Definition:One-Sided Polar|one-sided polar]] of $A$ in $\innerprod E F$.
Then $A^\circ \ne \O$. | Note that we have:
:$\innerprod x { {\mathbf 0}_F} = 0$ for all $x \in E$.
In particular:
:$\innerprod x { {\mathbf 0}_F} \le 1$ for all $x \in A$.
Hence we have ${\mathbf 0}_F \in A^\circ$.
Hence $A^\circ \ne \O$.
{{qed}}
[[Category:One-Sided Polars]]
banrxb2c7tm0lg4a7kgrrg0vysbpysc | One-Sided Polar is Non-Empty | https://proofwiki.org/wiki/One-Sided_Polar_is_Non-Empty | https://proofwiki.org/wiki/One-Sided_Polar_is_Non-Empty | [
"One-Sided Polars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Polar"
] | [
"Category:One-Sided Polars"
] |
proofwiki-22960 | One-Sided Polar is Convex | Let $\innerprod E F$ be a dual system over $\R$.
Let $A \subseteq E$ be non-empty.
Let $A^\odot$ be the one-sided polar of $A$ in $\innerprod E F$.
Then $A^\odot$ is convex. | Let $f, g \in A^\odot$ and $t \in \closedint 0 1$.
Then for each $x \in A$, we have:
{{begin-eqn}}
{{eqn | l = \innerprod x {t f + \paren {1 - t} g}
| r = t \innerprod x f + \paren {1 - t} \innerprod x g
| c = $\innerprod \cdot \cdot$ is bilinear
}}
{{eqn | o = \le
| r = t + \paren {1 - t}
}}
{{eqn | r = 1
}}
{{e... | Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $A \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $A^\odot$ be the [[Definition:One-Sided Polar|one-sided polar]] of $A$ in $\innerprod E F$.
Then $A^\odot$ is [[Definition:Convex Set (Vector Space)|convex]]. | Let $f, g \in A^\odot$ and $t \in \closedint 0 1$.
Then for each $x \in A$, we have:
{{begin-eqn}}
{{eqn | l = \innerprod x {t f + \paren {1 - t} g}
| r = t \innerprod x f + \paren {1 - t} \innerprod x g
| c = $\innerprod \cdot \cdot$ is [[Definition:Bilinear Mapping|bilinear]]
}}
{{eqn | o = \le
| r = t + \pare... | One-Sided Polar is Convex | https://proofwiki.org/wiki/One-Sided_Polar_is_Convex | https://proofwiki.org/wiki/One-Sided_Polar_is_Convex | [
"One-Sided Polars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Polar",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Bilinear Mapping",
"Definition:Convex Set (Vector Space)",
"Category:One-Sided Polars"
] |
proofwiki-22961 | One-Sided Prepolar is Convex | Let $\innerprod E F$ be a dual system over $\R$.
Let $B \subseteq F$ be non-empty.
Let $B_\odot$ be the one-sided prepolar of $B$ in $\innerprod E F$.
Then $B_\odot$ is convex. | Let $x, y \in B_\odot$ and $t \in \closedint 0 1$.
Then for each $f \in B$, we have:
{{begin-eqn}}
{{eqn | l = \innerprod {t x + \paren {1 - t} y} f
| r = t \innerprod x f + \paren {1 - t} \innerprod y f
| c = $\innerprod \cdot \cdot$ is bilinear
}}
{{eqn | o = \le
| r = t + \paren {1 - t}
}}
{{eqn | r = 1
}}
{{e... | Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $B \subseteq F$ be [[Definition:Non-Empty Set|non-empty]].
Let $B_\odot$ be the [[Definition:One-Sided Prepolar|one-sided prepolar]] of $B$ in $\innerprod E F$.
Then $B_\odot$ is [[Definition:Convex Set (Vector Space)|convex]]. | Let $x, y \in B_\odot$ and $t \in \closedint 0 1$.
Then for each $f \in B$, we have:
{{begin-eqn}}
{{eqn | l = \innerprod {t x + \paren {1 - t} y} f
| r = t \innerprod x f + \paren {1 - t} \innerprod y f
| c = $\innerprod \cdot \cdot$ is [[Definition:Bilinear Mapping|bilinear]]
}}
{{eqn | o = \le
| r = t + \pare... | One-Sided Prepolar is Convex | https://proofwiki.org/wiki/One-Sided_Prepolar_is_Convex | https://proofwiki.org/wiki/One-Sided_Prepolar_is_Convex | [
"One-Sided Prepolars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Prepolar",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Bilinear Mapping",
"Category:One-Sided Prepolars"
] |
proofwiki-22962 | One-Sided Prepolar is Closed in Weak Topology | Let $\innerprod E F$ be a dual system over $\R$.
Let $\map \sigma {E, F}$ be the weak topology on $E$ induced by $\innerprod E F$.
Let $B \subseteq F$ be non-empty.
Let $B_\odot$ be the one-sided prepolar of $B$ in $\innerprod E F$.
Then $B_\odot$ is $\map \sigma {E, F}$-closed. | We use Characterization of Closedness in terms of Nets.
We show that every convergent net valued in $B_\odot$ has its limit in $B_\odot$.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a convergent net valued in $B_\odot$ with limit $x \in E$.
Then for eac... | Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $\map \sigma {E, F}$ be the [[Definition:Weak Topology Induced by Dual System|weak topology on $E$]] induced by $\innerprod E F$.
Let $B \subseteq F$ be [[Definition:Non-Empty Set|non-empty]].
Let $B_\odot$ be the [[Definition:One-Sided ... | We use [[Characterization of Closedness in terms of Nets]].
We show that every [[Definition:Convergent Net|convergent net]] valued in $B_\odot$ has its [[Definition:Limit of Net|limit]] in $B_\odot$.
Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {x_\lambda}_{\lambda \ma... | One-Sided Prepolar is Closed in Weak Topology | https://proofwiki.org/wiki/One-Sided_Prepolar_is_Closed_in_Weak_Topology | https://proofwiki.org/wiki/One-Sided_Prepolar_is_Closed_in_Weak_Topology | [
"One-Sided Prepolars"
] | [
"Definition:Dual System",
"Definition:Weak Topology Induced by Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Prepolar",
"Definition:Closed Set"
] | [
"Characterization of Closedness in terms of Nets",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Definition:Directed Preordering",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Characterization of Convergence of Net in Weak Topology Induced by Dual System",
"Characterization of C... |
proofwiki-22963 | Bipolar Theorem/One-Sided Polar | Let $\GF \in \set {\R, \C}$.
Let $\innerprod X {X'}$ be a dual system over $\GF$.
Let $A \subseteq X$ be a non-empty set.
Let $\paren {A^\odot}_\odot$ be the one-sided bipolar of $A$.
Let $\map \sigma {X, X'}$ be the weak topology on $X$ induced by $\innerprod X {X'}$.
Let $\operatorname {conv}$ be the convex hull.
L... | From One-Sided Bipolar is Convex, $\paren {A^\odot}_\odot$ is convex.
From One-Sided Bipolar is Closed in Weak Topology, $\paren {A^\odot}_\odot$ is $\map \sigma {E, F}$-closed.
Let:
:$C = \map {\cl_\sigma} {\map {\operatorname {conv} } {A \cup \set { {\mathbf 0}_X} } }$
From One-Sided Bipolar Set contains Original Set... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod X {X'}$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $A \subseteq X$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $\paren {A^\odot}_\odot$ be the [[Definition:One-Sided Bipolar Set|one-sided bipolar]] of $A$.
Let $\map \sigma {X, X'}$ be the [[Def... | From [[One-Sided Bipolar is Convex]], $\paren {A^\odot}_\odot$ is [[Definition:Convex Set (Vector Space)|convex]].
From [[One-Sided Bipolar is Closed in Weak Topology]], $\paren {A^\odot}_\odot$ is [[Definition:Closed Set|$\map \sigma {E, F}$-closed]].
Let:
:$C = \map {\cl_\sigma} {\map {\operatorname {conv} } {A \cu... | Bipolar Theorem/One-Sided Polar | https://proofwiki.org/wiki/Bipolar_Theorem/One-Sided_Polar | https://proofwiki.org/wiki/Bipolar_Theorem/One-Sided_Polar | [
"Bipolar Theorem",
"One-Sided Polars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Bipolar Set",
"Definition:Weak Topology Induced by Dual System",
"Definition:Convex Hull",
"Definition:Closure (Topology)"
] | [
"One-Sided Bipolar is Convex",
"Definition:Convex Set (Vector Space)",
"One-Sided Bipolar is Closed in Weak Topology",
"Definition:Closed Set",
"One-Sided Bipolar Set contains Original Set and Origin",
"Closure of Convex Hull is Smallest Closed Convex Set containing Set",
"Hahn-Banach Separation Theorem... |
proofwiki-22964 | One-Sided Bipolar is Convex | Let $\innerprod E F$ be a dual system over $\R$.
Let $A \subseteq E$ be non-empty.
Let $\paren {A^\odot}_\odot$ be the one-sided bipolar of $A$ in $\innerprod E F$.
Then $\paren {A^\odot}_\odot$ is convex. | Let $A^\odot$ be the one-sided absolute polar of $A$.
From One-Sided Polar is Non-Empty, $A^\odot \ne \O$.
Hence from One-Sided Prepolar is Convex, $\paren {A^\odot}_\odot$ is convex.
{{qed}}
Category:One-Sided Bipolars
Category:Convex Sets (Vector Spaces)
rbxus0o8rcetxnt96jfp3z7zw2pfiui | Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $A \subseteq E$ be [[Definition:Non-Empty Set|non-empty]].
Let $\paren {A^\odot}_\odot$ be the [[Definition:One-Sided Bipolar Set|one-sided bipolar]] of $A$ in $\innerprod E F$.
Then $\paren {A^\odot}_\odot$ is [[Definition:Convex Set (... | Let $A^\odot$ be the [[Definition:One-Sided Polar|one-sided absolute polar]] of $A$.
From [[One-Sided Polar is Non-Empty]], $A^\odot \ne \O$.
Hence from [[One-Sided Prepolar is Convex]], $\paren {A^\odot}_\odot$ is [[Definition:Convex Set (Vector Space)|convex]].
{{qed}}
[[Category:One-Sided Bipolars]]
[[Category:Co... | One-Sided Bipolar is Convex | https://proofwiki.org/wiki/One-Sided_Bipolar_is_Convex | https://proofwiki.org/wiki/One-Sided_Bipolar_is_Convex | [
"Convex Sets (Vector Spaces)",
"One-Sided Bipolars",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Bipolar Set",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:One-Sided Polar",
"One-Sided Polar is Non-Empty",
"One-Sided Prepolar is Convex",
"Definition:Convex Set (Vector Space)",
"Category:One-Sided Bipolars",
"Category:Convex Sets (Vector Spaces)"
] |
proofwiki-22965 | Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 1 | :$I$ is a covariant functor | === Object Functor is Well-defined ===
Let $\tuple{D,f}$ be an object in $\paren{C \downarrow F}$.
By definition of functor under object comma category:
:$f$ is a morphism $f: C \to FD$ of $\mathbf C$
Hence:
:$f$ is a morphism $f: G* \to FD$ of $\mathbf C$
By definition of comma category:
:$\tuple{*, D, f}$ is an objec... | :$I$ is a [[Definition:Covariant Functor|covariant functor]] | === Object Functor is Well-defined ===
Let $\tuple{D,f}$ be an [[Definition:Object (Category Theory)|object]] in $\paren{C \downarrow F}$.
By definition of [[Definition:Functor Under Object Comma Category|functor under object comma category]]:
:$f$ is a [[Definition:Morphism (Category Theory)|morphism]] $f: C \to FD$... | Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 1 | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_1 | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_1 | [
"Functor Under Object Comma Category is Isomorphic to Comma Category"
] | [
"Definition:Functor/Covariant"
] | [
"Definition:Object (Category Theory)",
"Definition:Comma Category/Functor Under Object",
"Definition:Morphism",
"Definition:Morphism",
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Well-Defined",
"Definition:Object Functor",
"Definition:Morphism",
"De... |
proofwiki-22966 | One-Sided Bipolar Set contains Original Set and Origin | Let $\innerprod X {X'}$ be a dual system over $\R$.
Let $A \subseteq X$ be a non-empty set.
Let $\paren {A^\odot}_\odot$ be the one-sided bipolar of $A$.
Then:
:$A \cup \set { {\mathbf 0}_X} \subseteq \paren {A^\odot}_\odot$ | We have $x \in \paren {A^\odot}_\odot$ {{iff}}:
:$\innerprod x f \le 1$ for each $f \in A^\odot$.
We have that $f \in A^\odot$ {{iff}}:
:$\innerprod x f \le 1$ for each $x \in A$.
Hence for $x \in A$, we have:
:$\innerprod x f \le 1$ for each $f \in A^\odot$.
Hence $x \in \paren {A^\odot}_\odot$.
Hence $A \subseteq \p... | Let $\innerprod X {X'}$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $A \subseteq X$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $\paren {A^\odot}_\odot$ be the [[Definition:One-Sided Bipolar Set|one-sided bipolar]] of $A$.
Then:
:$A \cup \set { {\mathbf 0}_X} \subseteq \paren {A^\odot}_\odo... | We have $x \in \paren {A^\odot}_\odot$ {{iff}}:
:$\innerprod x f \le 1$ for each $f \in A^\odot$.
We have that $f \in A^\odot$ {{iff}}:
:$\innerprod x f \le 1$ for each $x \in A$.
Hence for $x \in A$, we have:
:$\innerprod x f \le 1$ for each $f \in A^\odot$.
Hence $x \in \paren {A^\odot}_\odot$.
Hence $A \subsete... | One-Sided Bipolar Set contains Original Set and Origin | https://proofwiki.org/wiki/One-Sided_Bipolar_Set_contains_Original_Set_and_Origin | https://proofwiki.org/wiki/One-Sided_Bipolar_Set_contains_Original_Set_and_Origin | [
"One-Sided Bipolars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Bipolar Set"
] | [
"One-Sided Bipolar Set is Non-Empty",
"Category:One-Sided Bipolars"
] |
proofwiki-22967 | Closure of Convex Hull is Smallest Closed Convex Set containing Set | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $\cl$ be the closure taken in $\struct {X, \tau}$.
Let $\operatorname {conv}$ denote convex hull.
Let $A \subseteq X$ be non-empty.
Then $\map \cl {\map {\operatorname {conv} } A}$ is $\subseteq$-smallest closed convex ... | From Closed Convex Hull in Topological Vector Space is Closed and Convex, $\map \cl {\map {\operatorname {conv} } A}$ is a closed convex set containing $A$.
Let $C$ be a closed convex set containing $A$.
From Convex Hull is Smallest Convex Set containing Set, we have:
:$\map {\operatorname {conv} } A \subseteq C$
From ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $\cl$ be the [[Definition:Closure (Topology)|closure]] taken in $\struct {X, \tau}$.
Let $\operatorname {conv}$ denote [[Definition:Convex Hull|convex hull]].
Let $A \subseteq ... | From [[Closed Convex Hull in Topological Vector Space is Closed and Convex]], $\map \cl {\map {\operatorname {conv} } A}$ is a [[Definition:Closed Set|closed]] [[Definition:Convex Set (Vector Space)|convex set]] containing $A$.
Let $C$ be a [[Definition:Closed Set|closed]] [[Definition:Convex Set (Vector Space)|convex... | Closure of Convex Hull is Smallest Closed Convex Set containing Set | https://proofwiki.org/wiki/Closure_of_Convex_Hull_is_Smallest_Closed_Convex_Set_containing_Set | https://proofwiki.org/wiki/Closure_of_Convex_Hull_is_Smallest_Closed_Convex_Set_containing_Set | [
"Closed Convex Hulls"
] | [
"Definition:Topological Vector Space",
"Definition:Closure (Topology)",
"Definition:Convex Hull",
"Definition:Non-Empty Set",
"Definition:Smallest Element",
"Definition:Closed Set",
"Definition:Convex Set (Vector Space)"
] | [
"Closed Convex Hull in Topological Vector Space is Closed and Convex",
"Definition:Closed Set",
"Definition:Convex Set (Vector Space)",
"Definition:Closed Set",
"Definition:Convex Set (Vector Space)",
"Convex Hull is Smallest Convex Set containing Set",
"Topological Closure of Subset is Subset of Topolo... |
proofwiki-22968 | Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 2 | :$J$ is a covariant functor | === Object Functor is Well-defined ===
By definition of category $\mathbf 1$:
:the only object in $\mathbf 1$ is $*$
By definition of comma category:
:every object in $\paren{G \downarrow F}$ is of the form $\tuple{*, D, f}$ where:
::$*$ is in $\mathbf 1$
::$D$ is an object of $\mathbf D$
::$f$ is a morphism $f:G* \to ... | :$J$ is a [[Definition:Covariant Functor|covariant functor]] | === Object Functor is Well-defined ===
By definition of [[Definition:One (Category)|category $\mathbf 1$]]:
:the only [[Definition:Object (Category Theory)|object]] in $\mathbf 1$ is $*$
By definition of [[Definition:Comma Category (General Form)|comma category]]:
:every [[Definition:Object (Category Theory)|object]... | Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 2 | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_2 | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_2 | [
"Functor Under Object Comma Category is Isomorphic to Comma Category"
] | [
"Definition:Functor/Covariant"
] | [
"Definition:One (Category)",
"Definition:Object (Category Theory)",
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Comma Category/... |
proofwiki-22969 | Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 3 | :$J \circ I = \operatorname{id}_{\paren{C \downarrow F} }$
where:
:$\operatorname{id}_{\paren{C \downarrow F} }$ denotes the identity functor on $\paren{C \downarrow F}$ | For each object $\tuple{D,f}$ in $\paren{C \downarrow F}$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J \circ I} \tuple{D, f}
| r = \map J {I \tuple{D, f} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J \tuple{*, D, f}
| c = Definition of $I$
}}
{{eqn | r = \tuple{D, f}
| c = Definition of $J... | :$J \circ I = \operatorname{id}_{\paren{C \downarrow F} }$
where:
:$\operatorname{id}_{\paren{C \downarrow F} }$ denotes the [[Definition:Identity Functor|identity functor]] on $\paren{C \downarrow F}$ | For each [[Definition:Object (Category Theory)|object]] $\tuple{D,f}$ in $\paren{C \downarrow F}$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J \circ I} \tuple{D, f}
| r = \map J {I \tuple{D, f} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J \tuple{*, D, f}
| c = Definition of $I$
}}
{{eqn | r = \... | Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 3 | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_3 | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_3 | [
"Functor Under Object Comma Category is Isomorphic to Comma Category"
] | [
"Definition:Identity Functor"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Functor Under Object Comma Category is Isomorphic to Comma Category"
] |
proofwiki-22970 | Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 4 | :$I \circ J = \operatorname{id}_{\paren{G \downarrow F} }$
where:
:$\operatorname{id}_{\paren{G \downarrow F} }$ denotes the identity functor on $\paren{G \downarrow F}$ | For each object $\tuple{*, D,f}$ in $\paren{G \downarrow F}$ we have:
{{begin-eqn}}
{{eqn | l = \paren{I \circ J} \tuple{*, D, f}
| r = \map I {J \tuple{*, D, f} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = I \tuple{D, f}
| c = Definition of $J$
}}
{{eqn | r = \tuple{*, D, f}
| c = Definit... | :$I \circ J = \operatorname{id}_{\paren{G \downarrow F} }$
where:
:$\operatorname{id}_{\paren{G \downarrow F} }$ denotes the [[Definition:Identity Functor|identity functor]] on $\paren{G \downarrow F}$ | For each [[Definition:Object (Category Theory)|object]] $\tuple{*, D,f}$ in $\paren{G \downarrow F}$ we have:
{{begin-eqn}}
{{eqn | l = \paren{I \circ J} \tuple{*, D, f}
| r = \map I {J \tuple{*, D, f} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = I \tuple{D, f}
| c = Definition of $J$
}}
{{eqn |... | Functor Under Object Comma Category is Isomorphic to Comma Category/Lemma 4 | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_4 | https://proofwiki.org/wiki/Functor_Under_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_4 | [
"Functor Under Object Comma Category is Isomorphic to Comma Category"
] | [
"Definition:Identity Functor"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Functor Under Object Comma Category is Isomorphic to Comma Category"
] |
proofwiki-22971 | Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 1 | :$I$ is a covariant functor | === Object Functor is Well-defined ===
Let $\tuple{E,g}$ be an object in $\paren{G \downarrow C}$.
By definition of functor over object comma category:
:$g$ is a morphism $g: GE \to C$ of $\mathbf C$
Hence:
:$g$ is a morphism $g: GE \to F*$ of $\mathbf C$
By definition of comma category:
:$\tuple{E, *, g}$ is an object... | :$I$ is a [[Definition:Covariant Functor|covariant functor]] | === Object Functor is Well-defined ===
Let $\tuple{E,g}$ be an [[Definition:Object (Category Theory)|object]] in $\paren{G \downarrow C}$.
By definition of [[Definition:Functor Over Object Comma Category|functor over object comma category]]:
:$g$ is a [[Definition:Morphism (Category Theory)|morphism]] $g: GE \to C$ o... | Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 1 | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_1 | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_1 | [
"Functor Over Object Comma Category is Isomorphic to Comma Category"
] | [
"Definition:Functor/Covariant"
] | [
"Definition:Object (Category Theory)",
"Definition:Comma Category/Functor Over Object",
"Definition:Morphism",
"Definition:Morphism",
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Well-Defined",
"Definition:Object Functor",
"Definition:Morphism",
"Def... |
proofwiki-22972 | Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 2 | :$J$ is a covariant functor | === Object Functor is Well-defined ===
By definition of category $\mathbf 1$:
:the only object in $\mathbf 1$ is $*$
By definition of comma category:
:every object in $\paren{G \downarrow F}$ is of the form $\tuple{E, *, g}$ where:
::$E$ is an object of $\mathbf D$
::$*$ is in $\mathbf 1$
::$g$ is a morphism $f:GE \to ... | :$J$ is a [[Definition:Covariant Functor|covariant functor]] | === Object Functor is Well-defined ===
By definition of [[Definition:One (Category)|category $\mathbf 1$]]:
:the only [[Definition:Object (Category Theory)|object]] in $\mathbf 1$ is $*$
By definition of [[Definition:Comma Category (General Form)|comma category]]:
:every [[Definition:Object (Category Theory)|object]... | Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 2 | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_2 | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_2 | [
"Functor Over Object Comma Category is Isomorphic to Comma Category"
] | [
"Definition:Functor/Covariant"
] | [
"Definition:One (Category)",
"Definition:Object (Category Theory)",
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Comma Category/... |
proofwiki-22973 | Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 3 | :$J \circ I = \operatorname{id}_{\paren{G \downarrow C}}$
where $\operatorname{id}_{\paren{G \downarrow C}}$ denotes the identity functor on $\paren{G \downarrow C}$ | For each object $\tuple{E,g}$ in $\paren{G \downarrow C}$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J \circ I} \tuple{E, g}
| r = \map J {I \tuple{E, g} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J \tuple{E, *, g}
| c = Definition of $I$
}}
{{eqn | r = \tuple{E, g}
| c = Definition of $J... | :$J \circ I = \operatorname{id}_{\paren{G \downarrow C}}$
where $\operatorname{id}_{\paren{G \downarrow C}}$ denotes the [[Definition:Identity Functor|identity functor]] on $\paren{G \downarrow C}$ | For each [[Definition:Object (Category Theory)|object]] $\tuple{E,g}$ in $\paren{G \downarrow C}$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J \circ I} \tuple{E, g}
| r = \map J {I \tuple{E, g} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J \tuple{E, *, g}
| c = Definition of $I$
}}
{{eqn | r = \... | Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 3 | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_3 | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_3 | [
"Functor Over Object Comma Category is Isomorphic to Comma Category"
] | [
"Definition:Identity Functor"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Functor Over Object Comma Category is Isomorphic to Comma Category"
] |
proofwiki-22974 | Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 4 | :$I \circ J = \operatorname{id}_{\paren{G \downarrow F}}$
where $\operatorname{id}_{\paren{G \downarrow F}}$ denotes the identity functor on $\paren{G \downarrow F}$ | For each object $\tuple{E, *, g}$ in $\paren{G \downarrow F}$ we have:
{{begin-eqn}}
{{eqn | l = \paren{I \circ J} \tuple{E, *, g}
| r = \map I {J \tuple{E, *, g} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = I \tuple{E, g}
| c = Definition of $J$
}}
{{eqn | r = \tuple{E, *, g}
| c = Defini... | :$I \circ J = \operatorname{id}_{\paren{G \downarrow F}}$
where $\operatorname{id}_{\paren{G \downarrow F}}$ denotes the [[Definition:Identity Functor|identity functor]] on $\paren{G \downarrow F}$ | For each [[Definition:Object (Category Theory)|object]] $\tuple{E, *, g}$ in $\paren{G \downarrow F}$ we have:
{{begin-eqn}}
{{eqn | l = \paren{I \circ J} \tuple{E, *, g}
| r = \map I {J \tuple{E, *, g} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = I \tuple{E, g}
| c = Definition of $J$
}}
{{eqn ... | Functor Over Object Comma Category is Isomorphic to Comma Category/Lemma 4 | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_4 | https://proofwiki.org/wiki/Functor_Over_Object_Comma_Category_is_Isomorphic_to_Comma_Category/Lemma_4 | [
"Functor Over Object Comma Category is Isomorphic to Comma Category"
] | [
"Definition:Identity Functor"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Functor Over Object Comma Category is Isomorphic to Comma Category"
] |
proofwiki-22975 | Slice Category is Isomorphic to Comma Category/Lemma 1 | :$I$ is a covariant functor | === Object Functor is Well-defined ===
Let $g$ be an object in $\mathbf C / C$.
By definition of slice category:
:$g$ is a morphism $g: \operatorname{dom} g \to C$ of $\mathbf C$
By definition of identity functor:
:$\map {\operatorname{id}_{\mathbf C} } {\operatorname{dom} g} = \operatorname{dom} g$
Hence:
:$g$ is a mo... | :$I$ is a [[Definition:Covariant Functor|covariant functor]] | === Object Functor is Well-defined ===
Let $g$ be an [[Definition:Object (Category Theory)|object]] in $\mathbf C / C$.
By definition of [[Definition:Slice Category|slice category]]:
:$g$ is a [[Definition:Morphism (Category Theory)|morphism]] $g: \operatorname{dom} g \to C$ of $\mathbf C$
By definition of [[Defini... | Slice Category is Isomorphic to Comma Category/Lemma 1 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Comma_Category/Lemma_1 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Comma_Category/Lemma_1 | [
"Slice Category is Isomorphic to Comma Category"
] | [
"Definition:Functor/Covariant"
] | [
"Definition:Object (Category Theory)",
"Definition:Slice Category",
"Definition:Morphism",
"Definition:Identity Functor",
"Definition:Morphism",
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Well-Defined",
"Definition:Object Functor",
"Definition:Morp... |
proofwiki-22976 | Slice Category is Isomorphic to Comma Category/Lemma 2 | :$J$ is a covariant functor | === Object Functor is Well-defined ===
By definition of category $\mathbf 1$:
:the only object in $\mathbf 1$ is $*$
By definition of comma category:
:every object in $\paren{\operatorname{id}_{\mathbf C} \downarrow F}$ is of the form $\tuple{E, *, g}$ where:
::$E$ is an object of $\mathbf D$
::$*$ is in $\mathbf 1$
::... | :$J$ is a [[Definition:Covariant Functor|covariant functor]] | === Object Functor is Well-defined ===
By definition of [[Definition:One (Category)|category $\mathbf 1$]]:
:the only [[Definition:Object (Category Theory)|object]] in $\mathbf 1$ is $*$
By definition of [[Definition:Comma Category (General Form)|comma category]]:
:every [[Definition:Object (Category Theory)|object]... | Slice Category is Isomorphic to Comma Category/Lemma 2 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Comma_Category/Lemma_2 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Comma_Category/Lemma_2 | [
"Slice Category is Isomorphic to Comma Category"
] | [
"Definition:Functor/Covariant"
] | [
"Definition:One (Category)",
"Definition:Object (Category Theory)",
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Slice Category"... |
proofwiki-22977 | Slice Category is Isomorphic to Comma Category/Lemma 3 | :$J \circ I = \operatorname{id}_{\mathbf C / C}$
where $\operatorname{id}_{\mathbf C / C}$ is the identity functor on $\mathbf C / C$ | For each object $g$ in $\mathbf C / C$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J \circ I} g
| r = \map J {I g }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J \tuple{\operatorname {dom} g, *, g}
| c = Definition of $I$
}}
{{eqn | r = g
| c = Definition of $J$
}}
{{eqn | r = \map {\operator... | :$J \circ I = \operatorname{id}_{\mathbf C / C}$
where $\operatorname{id}_{\mathbf C / C}$ is the [[Definition:Identity Functor|identity functor]] on $\mathbf C / C$ | For each [[Definition:Object (Category Theory)|object]] $g$ in $\mathbf C / C$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J \circ I} g
| r = \map J {I g }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J \tuple{\operatorname {dom} g, *, g}
| c = Definition of $I$
}}
{{eqn | r = g
| c = Definiti... | Slice Category is Isomorphic to Comma Category/Lemma 3 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Comma_Category/Lemma_3 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Comma_Category/Lemma_3 | [
"Slice Category is Isomorphic to Comma Category"
] | [
"Definition:Identity Functor"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Slice Category is Isomorphic to Comma Category"
] |
proofwiki-22978 | Slice Category is Isomorphic to Comma Category/Lemma 4 | :$I \circ J = \operatorname{id}_{\paren{\operatorname{id}_{\mathbf C} \downarrow F }}$
where $\operatorname{id}_{\paren{\operatorname{id}_{\mathbf C} \downarrow F }}$ is the identity functor on $\paren{\operatorname{id}_{\mathbf C} \downarrow F }$ | By definition of comma category:
:For each object $\tuple{E, *, g}$ in $\paren{\operatorname{id}_{\mathbf C} \downarrow F}$:
::$\text{(1)} \quad \operatorname{dom} g = E$
For each object $\tuple{E, *, g}$ in $\paren{\operatorname{id}_{\mathbf C} \downarrow F}$ we have:
{{begin-eqn}}
{{eqn | l = \paren{I \circ J} \tuple... | :$I \circ J = \operatorname{id}_{\paren{\operatorname{id}_{\mathbf C} \downarrow F }}$
where $\operatorname{id}_{\paren{\operatorname{id}_{\mathbf C} \downarrow F }}$ is the [[Definition:Identity Functor|identity functor]] on $\paren{\operatorname{id}_{\mathbf C} \downarrow F }$ | By definition of [[Definition:Comma Category (General Form)|comma category]]:
:For each [[Definition:Object (Category Theory)|object]] $\tuple{E, *, g}$ in $\paren{\operatorname{id}_{\mathbf C} \downarrow F}$:
::$\text{(1)} \quad \operatorname{dom} g = E$
For each [[Definition:Object (Category Theory)|object]] $\tupl... | Slice Category is Isomorphic to Comma Category/Lemma 4 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Comma_Category/Lemma_4 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Comma_Category/Lemma_4 | [
"Slice Category is Isomorphic to Comma Category"
] | [
"Definition:Identity Functor"
] | [
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Slice Category is Isomorphic to Comma Category"
] |
proofwiki-22979 | Coslice Category is Isomorphic to Comma Category/Lemma 1 | :$I$ is a covariant functor | === Object Functor is Well-defined ===
Let $f$ be an object in $C / \mathbf C$.
By definition of coslice category:
:$f$ is a morphism $f: C \to D$ of $\mathbf C$
By definition of identity functor:
:$\operatorname{id}_{\mathbf C} D = D$
Hence:
:$f$ is a morphism $f: G* \to \operatorname{id}_{\mathbf C} D$ of $\mathbf C$... | :$I$ is a [[Definition:Covariant Functor|covariant functor]] | === Object Functor is Well-defined ===
Let $f$ be an [[Definition:Object (Category Theory)|object]] in $C / \mathbf C$.
By definition of [[Definition:Coslice Category|coslice category]]:
:$f$ is a [[Definition:Morphism (Category Theory)|morphism]] $f: C \to D$ of $\mathbf C$
By definition of [[Definition:Identity F... | Coslice Category is Isomorphic to Comma Category/Lemma 1 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Comma_Category/Lemma_1 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Comma_Category/Lemma_1 | [
"Coslice Category is Isomorphic to Comma Category"
] | [
"Definition:Functor/Covariant"
] | [
"Definition:Object (Category Theory)",
"Definition:Coslice Category",
"Definition:Morphism",
"Definition:Identity Functor",
"Definition:Morphism",
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Well-Defined",
"Definition:Object Functor",
"Definition:Mo... |
proofwiki-22980 | Coslice Category is Isomorphic to Comma Category/Lemma 2 | :$J$ is a covariant functor | === Object Functor is Well-defined ===
By definition of category $\mathbf 1$:
:the only object in $\mathbf 1$ is $*$
By definition of comma category:
:every object in $\paren{G \downarrow \operatorname{id}_{\mathbf C}}$ is of the form $\tuple{*, D, f}$ where:
::$*$ is in $\mathbf 1$
::$D$ is an object of $\mathbf D$
::... | :$J$ is a [[Definition:Covariant Functor|covariant functor]] | === Object Functor is Well-defined ===
By definition of [[Definition:One (Category)|category $\mathbf 1$]]:
:the only [[Definition:Object (Category Theory)|object]] in $\mathbf 1$ is $*$
By definition of [[Definition:Comma Category (General Form)|comma category]]:
:every [[Definition:Object (Category Theory)|object]... | Coslice Category is Isomorphic to Comma Category/Lemma 2 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Comma_Category/Lemma_2 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Comma_Category/Lemma_2 | [
"Coslice Category is Isomorphic to Comma Category"
] | [
"Definition:Functor/Covariant"
] | [
"Definition:One (Category)",
"Definition:Object (Category Theory)",
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Identity Functor",
"Definition:Morphism",
"Definition:Coslice Category",
"D... |
proofwiki-22981 | Coslice Category is Isomorphic to Comma Category/Lemma 3 | :$J \circ I = \operatorname{id}_{C / \mathbf C}$
where $\operatorname{id}_{C / \mathbf C}$ is the identity functor on $C / \mathbf C$ | For each object $f$ in $C / \mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J \circ I} f
| r = \map J {I f }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J \tuple{*, \operatorname{cod} f, f}
| c = Definition of $I$
}}
{{eqn | r = f
| c = Definition of $J$
}}
{{eqn | r = \operatorname{id... | :$J \circ I = \operatorname{id}_{C / \mathbf C}$
where $\operatorname{id}_{C / \mathbf C}$ is the [[Definition:Identity Functor|identity functor]] on $C / \mathbf C$ | For each [[Definition:Object (Category Theory)|object]] $f$ in $C / \mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J \circ I} f
| r = \map J {I f }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J \tuple{*, \operatorname{cod} f, f}
| c = Definition of $I$
}}
{{eqn | r = f
| c = Definitio... | Coslice Category is Isomorphic to Comma Category/Lemma 3 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Comma_Category/Lemma_3 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Comma_Category/Lemma_3 | [
"Coslice Category is Isomorphic to Comma Category"
] | [
"Definition:Identity Functor"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Coslice Category is Isomorphic to Comma Category"
] |
proofwiki-22982 | Coslice Category is Isomorphic to Comma Category/Lemma 4 | :$I \circ J = \operatorname{id}_{\paren{G \downarrow \operatorname{id}_{\mathbf C} }}$
where $\operatorname{id}_{\paren{G \downarrow \operatorname{id}_{\mathbf C} }}$ is the identity functor on $\paren{G \downarrow \operatorname{id}_{\mathbf C} }$ | By definition of comma category:
:For each object $\tuple{*, D, f}$ in $\paren{G \downarrow \operatorname{id}_{\mathbf C} }$:
::$\text{(1)} \quad \operatorname{cod} f = D$
For each object $\tuple{*, D,f}$ in $\paren{G \downarrow \operatorname{id}_{\mathbf C} }$ we have:
{{begin-eqn}}
{{eqn | l = \paren{I \circ J} \tupl... | :$I \circ J = \operatorname{id}_{\paren{G \downarrow \operatorname{id}_{\mathbf C} }}$
where $\operatorname{id}_{\paren{G \downarrow \operatorname{id}_{\mathbf C} }}$ is the [[Definition:Identity Functor|identity functor]] on $\paren{G \downarrow \operatorname{id}_{\mathbf C} }$ | By definition of [[Definition:Comma Category (General Form)|comma category]]:
:For each [[Definition:Object (Category Theory)|object]] $\tuple{*, D, f}$ in $\paren{G \downarrow \operatorname{id}_{\mathbf C} }$:
::$\text{(1)} \quad \operatorname{cod} f = D$
For each [[Definition:Object (Category Theory)|object]] $\tup... | Coslice Category is Isomorphic to Comma Category/Lemma 4 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Comma_Category/Lemma_4 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Comma_Category/Lemma_4 | [
"Coslice Category is Isomorphic to Comma Category"
] | [
"Definition:Identity Functor"
] | [
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Coslice Category is Isomorphic to Comma Category"
] |
proofwiki-22983 | Morphism Category is Isomorphic to Comma Category/Lemma 1 | :$I$ is a covariant functor | === Object Functor is Well-defined ===
Let $f$ be an object in $\mathbf C^\to$.
By definition of morphism category:
:$f$ is a morphism $f: \operatorname{dom} f \to \operatorname{cod} f$ of $\mathbf C$
By definition of identity functor:
:$\map {\operatorname{id}_{\mathbf C} } {\operatorname{dom} f} = \operatorname{dom} ... | :$I$ is a [[Definition:Covariant Functor|covariant functor]] | === Object Functor is Well-defined ===
Let $f$ be an [[Definition:Object (Category Theory)|object]] in $\mathbf C^\to$.
By definition of [[Definition:Morphism Category|morphism category]]:
:$f$ is a [[Definition:Morphism (Category Theory)|morphism]] $f: \operatorname{dom} f \to \operatorname{cod} f$ of $\mathbf C$
... | Morphism Category is Isomorphic to Comma Category/Lemma 1 | https://proofwiki.org/wiki/Morphism_Category_is_Isomorphic_to_Comma_Category/Lemma_1 | https://proofwiki.org/wiki/Morphism_Category_is_Isomorphic_to_Comma_Category/Lemma_1 | [
"Morphism Category is Isomorphic to Comma Category"
] | [
"Definition:Functor/Covariant"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism Category",
"Definition:Morphism",
"Definition:Identity Functor",
"Definition:Morphism",
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Well-Defined",
"Definition:Object Functor",
"Definition:M... |
proofwiki-22984 | Morphism Category is Isomorphic to Comma Category/Lemma 2 | :$J$ is a covariant functor | === Object Functor is Well-defined ===
By definition of comma category:
:every object in $\paren{\operatorname{id}_{\mathbf C} \downarrow \operatorname{id}_{\mathbf C} }$ is of the form $\tuple{E, D, f}$:
:$f$ is a morphism $f : \operatorname{id}_{\mathbf C} E \to \operatorname{id}_{\mathbf C} D$ in $\mathbf C$
By defi... | :$J$ is a [[Definition:Covariant Functor|covariant functor]] | === Object Functor is Well-defined ===
By definition of [[Definition:Comma Category (General Form)|comma category]]:
:every [[Definition:Object (Category Theory)|object]] in $\paren{\operatorname{id}_{\mathbf C} \downarrow \operatorname{id}_{\mathbf C} }$ is of the form $\tuple{E, D, f}$:
:$f$ is a [[Definition:Morphi... | Morphism Category is Isomorphic to Comma Category/Lemma 2 | https://proofwiki.org/wiki/Morphism_Category_is_Isomorphic_to_Comma_Category/Lemma_2 | https://proofwiki.org/wiki/Morphism_Category_is_Isomorphic_to_Comma_Category/Lemma_2 | [
"Morphism Category is Isomorphic to Comma Category"
] | [
"Definition:Functor/Covariant"
] | [
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Identity Morphism",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Morphism Category",
"Definition:Object (Category Theory)",
"Definition:Well-Defined",
"D... |
proofwiki-22985 | Morphism Category is Isomorphic to Comma Category/Lemma 3 | :$J \circ I = \operatorname{id}_{\mathbf C^\to}$
where $\operatorname{id}_{\mathbf C^\to}$ is the identity functor on $\mathbf C^\to$ | For each object $g$ in $\mathbf C^\to$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J \circ I} g
| r = \map J {I g }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J \tuple{\operatorname {dom} g, \operatorname {cod} g, g}
| c = Definition of $I$
}}
{{eqn | r = g
| c = Definition of $J$
}}
{{eqn |... | :$J \circ I = \operatorname{id}_{\mathbf C^\to}$
where $\operatorname{id}_{\mathbf C^\to}$ is the [[Definition:Identity Functor|identity functor]] on $\mathbf C^\to$ | For each [[Definition:Object (Category Theory)|object]] $g$ in $\mathbf C^\to$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J \circ I} g
| r = \map J {I g }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J \tuple{\operatorname {dom} g, \operatorname {cod} g, g}
| c = Definition of $I$
}}
{{eqn | r = g
... | Morphism Category is Isomorphic to Comma Category/Lemma 3 | https://proofwiki.org/wiki/Morphism_Category_is_Isomorphic_to_Comma_Category/Lemma_3 | https://proofwiki.org/wiki/Morphism_Category_is_Isomorphic_to_Comma_Category/Lemma_3 | [
"Morphism Category is Isomorphic to Comma Category"
] | [
"Definition:Identity Functor"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Morphism Category is Isomorphic to Comma Category"
] |
proofwiki-22986 | Morphism Category is Isomorphic to Comma Category/Lemma 4 | :$I \circ J = \operatorname{id}_{\paren{\operatorname{id}_{\mathbf C} \downarrow \operatorname{id}_{\mathbf C} }}$
where $\operatorname{id}_{\paren{\operatorname{id}_{\mathbf C} \downarrow \operatorname{id}_{\mathbf C} }}$ is the identity functor on $\paren{\operatorname{id}_{\mathbf C} \downarrow \operatorname{id}_{\m... | By definition of comma category:
:For each object $\tuple{E, D, g}$ in $\paren{\operatorname{id}_{\mathbf C} \downarrow \paren{\operatorname{id}_{\mathbf C}}}$:
::$\text{(1)} \quad \operatorname{dom} g = E$
::$\text{(2)} \quad \operatorname{cod} g = D$
For each object $\tuple{E, D, g}$ in $\paren{\operatorname{id}_{\ma... | :$I \circ J = \operatorname{id}_{\paren{\operatorname{id}_{\mathbf C} \downarrow \operatorname{id}_{\mathbf C} }}$
where $\operatorname{id}_{\paren{\operatorname{id}_{\mathbf C} \downarrow \operatorname{id}_{\mathbf C} }}$ is the [[Definition:Identity Functor|identity functor]] on $\paren{\operatorname{id}_{\mathbf C} ... | By definition of [[Definition:Comma Category (General Form)|comma category]]:
:For each [[Definition:Object (Category Theory)|object]] $\tuple{E, D, g}$ in $\paren{\operatorname{id}_{\mathbf C} \downarrow \paren{\operatorname{id}_{\mathbf C}}}$:
::$\text{(1)} \quad \operatorname{dom} g = E$
::$\text{(2)} \quad \operato... | Morphism Category is Isomorphic to Comma Category/Lemma 4 | https://proofwiki.org/wiki/Morphism_Category_is_Isomorphic_to_Comma_Category/Lemma_4 | https://proofwiki.org/wiki/Morphism_Category_is_Isomorphic_to_Comma_Category/Lemma_4 | [
"Morphism Category is Isomorphic to Comma Category"
] | [
"Definition:Identity Functor"
] | [
"Definition:Comma Category/General Form",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Morphism Category is Isomorphic to Comma Category"
] |
proofwiki-22987 | One-Sided Prepolar is One-Sided Polar in Transpose | Let $\innerprod E F_{E \times F}$ be a dual system over $\GF$.
Let $\innerprod F E_{F \times E}$ be the transpose dual system.
Let $B \subseteq F$ be a non-empty set.
Let $B_\odot$ be the one-sided prepolar of $B$ taken in $\innerprod E F_{E \times F}$.
Let $B^{\odot^T}$ be the one-sided polar of $B$ taken in $\innerpr... | We have:
{{begin-eqn}}
{{eqn | l = B_\odot
| r = \set {x \in E :\innerprod x f_{E \times F} \le 1 \text { for all } f \in B}
}}
{{eqn | r = \set {x \in E : \innerprod f x_{F \times E} \le 1 \text { for all } f \in B}
}}
{{eqn | r = B^{\odot^T}
}}
{{end-eqn}}
{{qed}}
Category:One-Sided Prepolars
Category:One-Sided Pol... | Let $\innerprod E F_{E \times F}$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $\innerprod F E_{F \times E}$ be the [[Definition:Transpose Dual System|transpose dual system]].
Let $B \subseteq F$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $B_\odot$ be the [[Definition:One-Sided Prepolar|one-... | We have:
{{begin-eqn}}
{{eqn | l = B_\odot
| r = \set {x \in E :\innerprod x f_{E \times F} \le 1 \text { for all } f \in B}
}}
{{eqn | r = \set {x \in E : \innerprod f x_{F \times E} \le 1 \text { for all } f \in B}
}}
{{eqn | r = B^{\odot^T}
}}
{{end-eqn}}
{{qed}}
[[Category:One-Sided Prepolars]]
[[Category:One-Si... | One-Sided Prepolar is One-Sided Polar in Transpose | https://proofwiki.org/wiki/One-Sided_Prepolar_is_One-Sided_Polar_in_Transpose | https://proofwiki.org/wiki/One-Sided_Prepolar_is_One-Sided_Polar_in_Transpose | [
"One-Sided Prepolars",
"One-Sided Polars"
] | [
"Definition:Dual System",
"Definition:Transpose Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Prepolar",
"Definition:One-Sided Polar"
] | [
"Category:One-Sided Prepolars",
"Category:One-Sided Polars"
] |
proofwiki-22988 | One-Sided Polar is One-Sided Prepolar in Transpose | Let $\innerprod E F_{E \times F}$ be a dual system over $\GF$.
Let $\innerprod F E_{F \times E}$ be the transpose dual system.
Let $A \subseteq E$ be a non-empty set.
Let $A^\odot$ be the one-sided polar of $A$ taken in $\innerprod E F_{E \times F}$.
Let $A_{\odot^T}$ be the one-sided prepolar of $A$ taken in $\innerpr... | We have:
{{begin-eqn}}
{{eqn | l = A^\odot
| r = \set {f \in F :\innerprod x f_{E \times F} \le 1 \text { for all } x \in A}
}}
{{eqn | r = \set {f \in F : \innerprod f x_{F \times E} \le 1 \text { for all } x \in A}
}}
{{eqn | r = A_{\odot^T}
}}
{{end-eqn}}
{{qed}}
Category:One-Sided Polars
Category:One-Sided Prepol... | Let $\innerprod E F_{E \times F}$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $\innerprod F E_{F \times E}$ be the [[Definition:Transpose Dual System|transpose dual system]].
Let $A \subseteq E$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $A^\odot$ be the [[Definition:One-Sided Polar|one-sid... | We have:
{{begin-eqn}}
{{eqn | l = A^\odot
| r = \set {f \in F :\innerprod x f_{E \times F} \le 1 \text { for all } x \in A}
}}
{{eqn | r = \set {f \in F : \innerprod f x_{F \times E} \le 1 \text { for all } x \in A}
}}
{{eqn | r = A_{\odot^T}
}}
{{end-eqn}}
{{qed}}
[[Category:One-Sided Polars]]
[[Category:One-Sided... | One-Sided Polar is One-Sided Prepolar in Transpose | https://proofwiki.org/wiki/One-Sided_Polar_is_One-Sided_Prepolar_in_Transpose | https://proofwiki.org/wiki/One-Sided_Polar_is_One-Sided_Prepolar_in_Transpose | [
"One-Sided Polars",
"One-Sided Prepolars"
] | [
"Definition:Dual System",
"Definition:Transpose Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Polar",
"Definition:One-Sided Prepolar"
] | [
"Category:One-Sided Polars",
"Category:One-Sided Prepolars"
] |
proofwiki-22989 | Banach-Dieudonné Theorem | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_{X^\ast}^-$ be the closed unit ball in $X^\ast$.
Let $w^\ast$ be the weak-$\ast$ topology in ... | We first consider the case $\GF = \R$.
Note that the conclusion is entirely independent of whether $\R$ or $\C$ is used, so this is unproblematic.
Throughout we work in the dual system $\innerprod X {X^\ast}$ over $\R$, where:
:$\innerprod x f = \map f x$ for each $\tuple {x, f} \in X \times X^\ast$.
We are assured tha... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_{X^\ast}^-$ be the [[Definiti... | We first consider the case $\GF = \R$.
Note that the conclusion is entirely independent of whether $\R$ or $\C$ is used, so this is unproblematic.
Throughout we work in the [[Definition:Dual System|dual system]] $\innerprod X {X^\ast}$ over $\R$, where:
:$\innerprod x f = \map f x$ for each $\tuple {x, f} \in X \time... | Banach-Dieudonné Theorem | https://proofwiki.org/wiki/Banach-Dieudonné_Theorem | https://proofwiki.org/wiki/Banach-Dieudonné_Theorem | [
"Banach-Dieudonné Theorem",
"Weak-* Topologies",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Banach Space",
"Definition:Normed Dual Space",
"Definition:Closed Unit Ball",
"Definition:Weak-* Topology",
"Definition:Convex Set (Vector Space)",
"Definition:Closed Set",
"Definition:Closed Set"
] | [
"Definition:Dual System",
"Definition:Dual System",
"Normed Vector Space with Normed Dual Space is Dual System",
"Definition:Absolute Polar",
"Definition:One-Sided Polar",
"Definition:Dual System",
"Definition:Closed Set",
"Closed Unit Ball in Normed Dual Space is Weak-* Closed",
"Definition:Closed ... |
proofwiki-22990 | One-Sided Polar of Symmetric Set is equal to Absolute Polar | Let $\innerprod E F$ be a dual system over $\R$.
Let $A \subseteq E$ be a symmetric set.
Let $A^\odot$ be the one-sided polar of $A$.
Let $A^\circ$ be the absolute polar of $A$.
Then $A^\odot = A^\circ$. | Let $f \in A^\circ$.
Then for each $x \in A$, we have:
:$\cmod {\innerprod x f} \le 1$
and in particular:
:$\innerprod x f \le 1$
Hence $f \in A^\odot$.
Hence we have:
:$A^\circ \subseteq A^\odot$
We now just need to show that:
:$A^\odot \subseteq A^\circ$
Let $f \in A^\odot$.
Let $x \in A$.
Then:
:$\innerprod x f \le ... | Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $A \subseteq E$ be a [[Definition:Symmetric Set|symmetric set]].
Let $A^\odot$ be the [[Definition:One-Sided Polar|one-sided polar]] of $A$.
Let $A^\circ$ be the [[Definition:Absolute Polar|absolute polar]] of $A$.
Then $A^\odot = A^\... | Let $f \in A^\circ$.
Then for each $x \in A$, we have:
:$\cmod {\innerprod x f} \le 1$
and in particular:
:$\innerprod x f \le 1$
Hence $f \in A^\odot$.
Hence we have:
:$A^\circ \subseteq A^\odot$
We now just need to show that:
:$A^\odot \subseteq A^\circ$
Let $f \in A^\odot$.
Let $x \in A$.
Then:
:$\innerprod ... | One-Sided Polar of Symmetric Set is equal to Absolute Polar | https://proofwiki.org/wiki/One-Sided_Polar_of_Symmetric_Set_is_equal_to_Absolute_Polar | https://proofwiki.org/wiki/One-Sided_Polar_of_Symmetric_Set_is_equal_to_Absolute_Polar | [
"One-Sided Polars",
"Absolute Polars",
"One-Sided Polars",
"Absolute Polars"
] | [
"Definition:Dual System",
"Definition:Symmetric Set",
"Definition:One-Sided Polar",
"Definition:Absolute Polar"
] | [
"Definition:Symmetric Set",
"Definition:Bilinear Mapping",
"Category:One-Sided Polars",
"Category:Absolute Polars"
] |
proofwiki-22991 | Normed Vector Space with Normed Dual Space is Dual System | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Define $\innerprod \cdot \cdot : X \times X^\ast \to \GF$ by:
:$\innerprod x f = \map f x$ for each $\tu... | Let $X^\#$ be the algebraic dual of $X$.
From Normed Dual Space is Banach Space, $X^\ast$ is a vector subspace of $X^\#$.
From Normed Dual Space Separates Points, $X^\ast$ separates points.
Hence from Vector Space with Subspace of Algebraic Dual Separating Points is Dual System, $\tuple {X, X^\ast, \innerprod \cdot \cd... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Define $\innerprod \cdot \cdot : ... | Let $X^\#$ be the [[Definition:Algebraic Dual|algebraic dual]] of $X$.
From [[Normed Dual Space is Banach Space]], $X^\ast$ is a [[Definition:Vector Subspace|vector subspace]] of $X^\#$.
From [[Normed Dual Space Separates Points]], $X^\ast$ [[Definition:Mappings Separating Points|separates points]].
Hence from [[Vec... | Normed Vector Space with Normed Dual Space is Dual System | https://proofwiki.org/wiki/Normed_Vector_Space_with_Normed_Dual_Space_is_Dual_System | https://proofwiki.org/wiki/Normed_Vector_Space_with_Normed_Dual_Space_is_Dual_System | [
"Normed Vector Spaces",
"Dual Systems"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Dual System"
] | [
"Definition:Algebraic Dual",
"Normed Dual Space is Banach Space",
"Definition:Vector Subspace",
"Normed Dual Space Separates Points",
"Definition:Mappings Separating Points",
"Vector Space with Subspace of Algebraic Dual Separating Points is Dual System",
"Definition:Dual System",
"Category:Normed Vec... |
proofwiki-22992 | Sphere is not Developable Surface | The sphere is ''not'' a developable surface. | {{ProofWanted|Little formal work on curvature has been established on {{ProofWiki}}, so this is being left open while some of that work has been accomplished. Should be a result Gaussian Curvature of Sphere equals Curvature of Great Circle or some such}} | The [[Definition:Sphere (Geometry)|sphere]] is ''not'' a [[Definition:Developable Surface|developable surface]]. | {{ProofWanted|Little formal work on curvature has been established on {{ProofWiki}}, so this is being left open while some of that work has been accomplished. Should be a result [[Gaussian Curvature of Sphere equals Curvature of Great Circle]] or some such}} | Sphere is not Developable Surface | https://proofwiki.org/wiki/Sphere_is_not_Developable_Surface | https://proofwiki.org/wiki/Sphere_is_not_Developable_Surface | [
"Spheres",
"Developable Surfaces"
] | [
"Definition:Sphere/Geometry",
"Definition:Developable Surface"
] | [
"Gaussian Curvature of Sphere equals Curvature of Great Circle"
] |
proofwiki-22993 | One-Sided Polar is Closed in Weak-* Topology | Let $\innerprod E F$ be a dual system over $\R$.
Let $\map \sigma {F, E}$ be the weak-$\ast$ topology on $F$ induced by $\innerprod E F$.
Let $A \subseteq E$ be a non-empty set.
Let $A^\odot$ be the one-sided polar.
Then $A^\odot$ is $\map \sigma {F, E}$-closed. | We use Characterization of Closedness in terms of Nets.
We show that every convergent net valued in $A^\odot$ has its limit in $A^\odot$.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {f_\lambda}_{\lambda \mathop \in \Lambda}$ be a convergent net valued in $A^\odot$ with limit $f \in F$.
Then for ea... | Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $\map \sigma {F, E}$ be the [[Definition:Weak-* Topology Induced by Dual System|weak-$\ast$ topology]] on $F$ induced by $\innerprod E F$.
Let $A \subseteq E$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $A^\odot$ be the [[Defini... | We use [[Characterization of Closedness in terms of Nets]].
We show that every [[Definition:Convergent Net|convergent net]] valued in $A^\odot$ has its [[Definition:Limit of Net|limit]] in $A^\odot$.
Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {f_\lambda}_{\lambda \ma... | One-Sided Polar is Closed in Weak-* Topology | https://proofwiki.org/wiki/One-Sided_Polar_is_Closed_in_Weak-*_Topology | https://proofwiki.org/wiki/One-Sided_Polar_is_Closed_in_Weak-*_Topology | [
"One-Sided Polars",
"Weak Topologies Induced by Dual Systems"
] | [
"Definition:Dual System",
"Definition:Weak-* Topology Induced by Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Polar",
"Definition:Closed Set"
] | [
"Characterization of Closedness in terms of Nets",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Definition:Directed Preordering",
"Definition:Convergent Net",
"Definition:Limit of Net",
"Characterization of Convergence of Net in Weak-* Topology Induced by Dual System",
"Characterization of... |
proofwiki-22994 | Closed Unit Ball in Normed Vector Space is Balanced | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $B_X^-$ be the closed unit ball of $\struct {X, \norm {\, \cdot \,}_X}$.
Then $B_X^-$ is balanced. | Let $x \in B_X^-$.
Let $\lambda \in \GF$ be such that $\cmod \lambda \le 1$.
We have:
{{begin-eqn}}
{{eqn | l = \norm {\lambda x}_X
| r = \cmod \lambda \norm x_X
| c = {{NormAxiomVector|2}}
}}
{{eqn | o = \le
| r = \norm x_X
}}
{{eqn | o = \le
| r = 1
}}
{{end-eqn}}
{{qed}}
Category:Balanced Sets
Category:Nor... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $B_X^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Then $B_X^-$ is [[Definition:Balanced Set|balanced]]. | Let $x \in B_X^-$.
Let $\lambda \in \GF$ be such that $\cmod \lambda \le 1$.
We have:
{{begin-eqn}}
{{eqn | l = \norm {\lambda x}_X
| r = \cmod \lambda \norm x_X
| c = {{NormAxiomVector|2}}
}}
{{eqn | o = \le
| r = \norm x_X
}}
{{eqn | o = \le
| r = 1
}}
{{end-eqn}}
{{qed}}
[[Category:Balanced Sets]]
[[Cat... | Closed Unit Ball in Normed Vector Space is Balanced | https://proofwiki.org/wiki/Closed_Unit_Ball_in_Normed_Vector_Space_is_Balanced | https://proofwiki.org/wiki/Closed_Unit_Ball_in_Normed_Vector_Space_is_Balanced | [
"Balanced Sets",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Unit Ball",
"Definition:Balanced Set"
] | [
"Category:Balanced Sets",
"Category:Normed Vector Spaces"
] |
proofwiki-22995 | Absolute Polar of Closed Unit Ball in Normed Vector Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of \struct {X, \norm {\, \cdot \,}_X}$.
Let $B_X^-$ be the closed unit ball of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_{X^\ast}^-... | We have:
{{begin-eqn}}
{{eqn | l = \paren {B_X^-}^\circ
| r = \set {f \in X^\ast : \cmod {\map f x} \le 1 \text { for each } x \in B_X^-}
}}
{{eqn | r = \set {f \in X^\ast : \norm f_{X^\ast} \le 1}
| c = {{Defof|Norm on Bounded Linear Functional}}
}}
{{eqn | r = B_{X^\ast}^-
}}
{{end-eqn}}
{{qed}}
Category:Absolute... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of \struct {X, \norm {\, \cdot \,}_X}$.
Let $B_X^-$ be the [[De... | We have:
{{begin-eqn}}
{{eqn | l = \paren {B_X^-}^\circ
| r = \set {f \in X^\ast : \cmod {\map f x} \le 1 \text { for each } x \in B_X^-}
}}
{{eqn | r = \set {f \in X^\ast : \norm f_{X^\ast} \le 1}
| c = {{Defof|Norm on Bounded Linear Functional}}
}}
{{eqn | r = B_{X^\ast}^-
}}
{{end-eqn}}
{{qed}}
[[Category:Absol... | Absolute Polar of Closed Unit Ball in Normed Vector Space | https://proofwiki.org/wiki/Absolute_Polar_of_Closed_Unit_Ball_in_Normed_Vector_Space | https://proofwiki.org/wiki/Absolute_Polar_of_Closed_Unit_Ball_in_Normed_Vector_Space | [
"Absolute Polars",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Closed Unit Ball",
"Definition:Closed Unit Ball",
"Definition:Absolute Polar",
"Definition:Dual System"
] | [
"Category:Absolute Polars",
"Category:Normed Vector Spaces"
] |
proofwiki-22996 | One-Sided Polar of Closed Unit Ball in Normed Vector Space | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\R$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of \struct {X, \norm {\, \cdot \,}_X}$.
Let $B_X^-$ be the closed unit ball of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_{X^\ast}^-$ be the closed unit ball of $... | Let $\paren {B_X^-}^\circ$ be the absolute polar of $B_X^-$.
From Closed Unit Ball in Normed Vector Space is Balanced, we have that $B_X^-$ is balanced.
Hence from One-Sided Polar of Symmetric Set is equal to Absolute Polar and Balanced Set in Vector Space is Symmetric, we have:
:$\paren {B_X^-}^\odot = \paren {B_X^-}^... | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\R$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of \struct {X, \norm {\, \cdot \,}_X}$.
Let $B_X^-$ be the [[Definition:Closed Unit Ball|close... | Let $\paren {B_X^-}^\circ$ be the [[Definition:Absolute Polar|absolute polar]] of $B_X^-$.
From [[Closed Unit Ball in Normed Vector Space is Balanced]], we have that $B_X^-$ is [[Definition:Balanced Set|balanced]].
Hence from [[One-Sided Polar of Symmetric Set is equal to Absolute Polar]] and [[Balanced Set in Vector... | One-Sided Polar of Closed Unit Ball in Normed Vector Space | https://proofwiki.org/wiki/One-Sided_Polar_of_Closed_Unit_Ball_in_Normed_Vector_Space | https://proofwiki.org/wiki/One-Sided_Polar_of_Closed_Unit_Ball_in_Normed_Vector_Space | [
"One-Sided Polars",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Closed Unit Ball",
"Definition:Closed Unit Ball",
"Definition:One-Sided Polar",
"Definition:Dual System"
] | [
"Definition:Absolute Polar",
"Closed Unit Ball in Normed Vector Space is Balanced",
"Definition:Balanced Set",
"One-Sided Polar of Symmetric Set is equal to Absolute Polar",
"Balanced Set in Vector Space is Symmetric",
"Absolute Polar of Closed Unit Ball in Normed Vector Space",
"Category:One-Sided Pola... |
proofwiki-22997 | Absolute Prepolar of Closed Unit Ball in Normed Dual Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_X^-$ be the closed unit ball of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_{X^\ast}^... | We have:
{{begin-eqn}}
{{eqn | l = \paren {B_{X^\ast}^-}_\circ
| r = \set {x \in X : \cmod {\map f x} \le 1 \text { for all } f \in B_{X^\ast}^-}
}}
{{eqn | r = \set {x \in X : \sup_{f \mathop \in B_{X^\ast}^-} \cmod {\map f x} \le 1}
}}
{{eqn | r = \set {x \in X : \norm x_X \le 1}
| c = Norm in terms of Normed Dua... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $B_X^-$ be the [[D... | We have:
{{begin-eqn}}
{{eqn | l = \paren {B_{X^\ast}^-}_\circ
| r = \set {x \in X : \cmod {\map f x} \le 1 \text { for all } f \in B_{X^\ast}^-}
}}
{{eqn | r = \set {x \in X : \sup_{f \mathop \in B_{X^\ast}^-} \cmod {\map f x} \le 1}
}}
{{eqn | r = \set {x \in X : \norm x_X \le 1}
| c = [[Norm in terms of Normed D... | Absolute Prepolar of Closed Unit Ball in Normed Dual Space | https://proofwiki.org/wiki/Absolute_Prepolar_of_Closed_Unit_Ball_in_Normed_Dual_Space | https://proofwiki.org/wiki/Absolute_Prepolar_of_Closed_Unit_Ball_in_Normed_Dual_Space | [
"Absolute Prepolars",
"Normed Dual Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Closed Unit Ball",
"Definition:Closed Unit Ball",
"Definition:Absolute Prepolar",
"Definition:Dual System"
] | [
"Norm in terms of Normed Dual Space",
"Category:Absolute Prepolars",
"Category:Normed Dual Spaces"
] |
proofwiki-22998 | Real Matrix is Diagonalizable if Roots of Characteristic Equation are Real and Distinct | Let $\mathbf A$ be a square matrix of order $n$ whose entries are real.
Let the $n$ roots of the characteristic equation of $\mathbf A$ be real and distinct.
Then $\mathbf A$ is diagonalizable in $\mathbb R^n$. | From Number of Roots of Polynomial With Complex Coefficients Equals Degree of Polynomial, $n$ distinct roots of a degree $n$ characteristic polynomial must have algebraic multipicity of $1$.
Since Algebraic Multiplicity is not Less than Geometric Multiplicity, the geometric multiplicity of each eigenvalue is $1$.
Thus,... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order]] $n$ whose [[Definition:Matrix Entry|entries]] are [[Definition:Real Number|real]].
Let the $n$ [[Definition:Root of Equation|roots]] of the [[Definition:Characteristic Equation of Matrix|characteristic equati... | From [[Number of Roots of Polynomial With Complex Coefficients Equals Degree of Polynomial]], $n$ distinct [[Definition:Root of Polynomial|roots]] of a [[Definition:Degree of Polynomial|degree]] $n$ [[Definition:Characteristic Polynomial of Matrix|characteristic polynomial]] must have [[Definition:Algebraic Multiplicit... | Real Matrix is Diagonalizable if Roots of Characteristic Equation are Real and Distinct | https://proofwiki.org/wiki/Real_Matrix_is_Diagonalizable_if_Roots_of_Characteristic_Equation_are_Real_and_Distinct | https://proofwiki.org/wiki/Real_Matrix_is_Diagonalizable_if_Roots_of_Characteristic_Equation_are_Real_and_Distinct | [
"Diagonalizable Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix/Element",
"Definition:Real Number",
"Definition:Root of Equation",
"Definition:Characteristic Equation of Matrix",
"Definition:Real Number",
"Definition:Distinct/Plural",
"Definition:Diagonalizable Matrix"... | [
"Number of Roots of Polynomial With Complex Coefficients Equals Degree of Polynomial",
"Definition:Root of Polynomial",
"Definition:Degree of Polynomial",
"Definition:Characteristic Polynomial of Matrix",
"Definition:Algebraic Multiplicity",
"Algebraic Multiplicity is not Less than Geometric Multiplicity"... |
proofwiki-22999 | Powers of Diagonalizable Matrix | Let $\mathbf A$ be a square matrix which is diagonalizable matrix.
Let $\mathbf D$ be the diagonal matrix which is the result of the equation:
:$\mathbf D = \mathbf X^{-1} \mathbf A \mathbf X$
for some nonsingular matrix $\mathbf X$.
Then for all $n \in \Z_{\ge 0}$:
:$\mathbf A^n = \mathbf X \mathbf D^n \mathbf X^{-1}$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\mathbf A^n = \mathbf X \mathbf D^n \mathbf X^{-1}$
Let $\mathbf A$ be of dimension $m$ where $m \in \Z_{\ge 1}$.
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = \mathbf A^0
| r = \mathbf I_m
| c = where $\ma... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] which is [[Definition:Diagonalizable Matrix|diagonalizable matrix]].
Let $\mathbf D$ be the [[Definition:Diagonal Matrix|diagonal matrix]] which is the result of the [[Definition:Equation|equation]]:
:$\mathbf D = \mathbf X^{-1} \mathbf A \mathbf X$
for s... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\mathbf A^n = \mathbf X \mathbf D^n \mathbf X^{-1}$
Let $\mathbf A$ be of [[Definition:Dimension of Square Matrix|dimension]] $m$ where $m \in \Z_{\ge 1... | Powers of Diagonalizable Matrix | https://proofwiki.org/wiki/Powers_of_Diagonalizable_Matrix | https://proofwiki.org/wiki/Powers_of_Diagonalizable_Matrix | [
"Powers of Diagonalizable Matrix",
"Diagonalizable Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Diagonalizable Matrix",
"Definition:Diagonal Matrix",
"Definition:Equation",
"Definition:Nonsingular Matrix"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Matrix/Square Matrix/Order",
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix/Order",
"Principle of Mathematical Induction"
] |
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