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-23000 | Existence of Continuous Linear Functional separating Point in Hausdorff Locally Convex Space from Convex Set with Non-Empty Interior | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\GF$.
Let $X^\ast$ be the topological dual space of $\struct {X, \PP}$.
Let $C \subseteq X$ be a convex set with non-empty interior.
Let $\Int C$ be the interior of $C$.
Let $x_0 \in X \setminus C$.
Then there exists $f \in ... | Since $x \in X \setminus C$, we also have $x \in X \setminus \Int C$.
From Interior of Convex Set in Topological Vector Space is Convex, $\Int C$ is convex.
From Hahn-Banach Separation Theorem: Hausdorff Locally Convex Space: Complex Case: Open Convex Set and Convex Set applied to $\set {x_0}$ and $\Int C$, there exis... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Hausdorff Locally Convex Space|Hausdorff locally convex space]] over $\GF$.
Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $\struct {X, \PP}$.
Let $C \subseteq X$ be a [[Definition:Convex Set (Vector Space)|con... | Since $x \in X \setminus C$, we also have $x \in X \setminus \Int C$.
From [[Interior of Convex Set in Topological Vector Space is Convex]], $\Int C$ is [[Definition:Convex Set (Vector Space)|convex]].
From [[Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Complex Case/Open Convex Set and Convex Set|Ha... | Existence of Continuous Linear Functional separating Point in Hausdorff Locally Convex Space from Convex Set with Non-Empty Interior | https://proofwiki.org/wiki/Existence_of_Continuous_Linear_Functional_separating_Point_in_Hausdorff_Locally_Convex_Space_from_Convex_Set_with_Non-Empty_Interior | https://proofwiki.org/wiki/Existence_of_Continuous_Linear_Functional_separating_Point_in_Hausdorff_Locally_Convex_Space_from_Convex_Set_with_Non-Empty_Interior | [
"Hausdorff Locally Convex Spaces",
"Locally Convex Spaces",
"Locally Convex Spaces"
] | [
"Definition:Locally Convex Space/Hausdorff",
"Definition:Topological Dual Space",
"Definition:Convex Set (Vector Space)",
"Definition:Non-Empty Set",
"Definition:Interior (Topology)",
"Definition:Interior (Topology)"
] | [
"Interior of Convex Set in Topological Vector Space is Convex",
"Definition:Convex Set (Vector Space)",
"Hahn-Banach Separation Theorem/Hausdorff Locally Convex Space/Complex Case/Open Convex Set and Convex Set",
"Dilation of Open Set in Topological Vector Space is Open",
"Definition:Open Set",
"Sum of Se... |
proofwiki-23001 | Dilation of Closed Unit Ball in Normed Vector Space Decomposes as Sum of Dilations | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $\lambda, \mu > 0$.
Let $B_X^-$ be the closed unit ball in $\struct {X, \norm {\, \cdot \,}_X}$.
Then:
:$\paren {\lambda + \mu} B_X^- = \lambda B_X^- + \mu B_X^-$ | Let $y \in \paren {\lambda + \mu} B_X^-$.
Then there exists $x \in B_X^-$ such that $y = \paren {\lambda + \mu} x$.
Write:
:$y = \lambda x + \mu x \in \lambda B_X^- + \mu B_X^-$
We therefore have that $\paren {\lambda + \mu} B_X^- \subseteq \lambda B_X^- + \mu B_X^-$.
Now let $y \in \lambda B_X^- + \mu B_X^-$.
Hence t... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\lambda, \mu > 0$.
Let $B_X^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $\struct {X, \norm {\, \cdot \,}_X}$.
Then:
:$\paren {\lambda + \mu} B_X^- = \... | Let $y \in \paren {\lambda + \mu} B_X^-$.
Then there exists $x \in B_X^-$ such that $y = \paren {\lambda + \mu} x$.
Write:
:$y = \lambda x + \mu x \in \lambda B_X^- + \mu B_X^-$
We therefore have that $\paren {\lambda + \mu} B_X^- \subseteq \lambda B_X^- + \mu B_X^-$.
Now let $y \in \lambda B_X^- + \mu B_X^-$.
H... | Dilation of Closed Unit Ball in Normed Vector Space Decomposes as Sum of Dilations | https://proofwiki.org/wiki/Dilation_of_Closed_Unit_Ball_in_Normed_Vector_Space_Decomposes_as_Sum_of_Dilations | https://proofwiki.org/wiki/Dilation_of_Closed_Unit_Ball_in_Normed_Vector_Space_Decomposes_as_Sum_of_Dilations | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Unit Ball"
] | [
"Category:Normed Vector Spaces"
] |
proofwiki-23002 | Linear First Order ODE/y' - y = 3 | The linear first order ODE:
:$(1): \quad \dfrac {\d y} {\d x} - y = 3$
has the general solution:
:$y = C e^x - 3$ | $(1)$ is a linear first order ODE in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x = -1$
:$\map Q x = 3$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = -\int \rd x
| c =
}}
{{eqn | r = -x
| c =
}}
{{eqn | ll= \leadsto
| l = e^{\int P \rd x}
| r = ... | The [[Definition:Linear First Order ODE|linear first order ODE]]:
:$(1): \quad \dfrac {\d y} {\d x} - y = 3$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = C e^x - 3$ | $(1)$ is a [[Definition:Linear First Order ODE|linear first order ODE]] in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map P x = -1$
:$\map Q x = 3$
Thus:
{{begin-eqn}}
{{eqn | l = \int \map P x \rd x
| r = -\int \rd x
| c =
}}
{{eqn | r = -x
| c =
}}
{{eqn | ll= \leadsto
... | Linear First Order ODE/y' - y = 3 | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_-_y_=_3 | https://proofwiki.org/wiki/Linear_First_Order_ODE/y'_-_y_=_3 | [
"Examples of Linear First Order ODEs"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor",
"Primitive of Exponential of a x"
] |
proofwiki-23003 | Linear Second Order ODE/y'' - 2 y' - 3 y = e^(-x) | The second order ODE:
:$(1): \quad y' ' - 2 y' - 3 y = e^{-x}$
has the general solution:
:$y = C_1 e^{3 x} + C_2 e^{-x} - \dfrac {x e^{-x} } 4$ | It can be seen that $(1)$ is a nonhomogeneous linear second order ODE in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = -3$
:$\map R x = e^{-x}$
First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
:$y' ' - 2 y' - 3 y = 0$
From Linear Second Or... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' - 2 y' - 3 y = e^{-x}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = C_1 e^{3 x} + C_2 e^{-x} - \dfrac {x e^{-x} } 4$ | It can be seen that $(1)$ is a [[Definition:Nonhomogeneous Linear Second Order ODE|nonhomogeneous linear second order ODE]] in the form:
:$y' ' + p y' + q y = \map R x$
where:
:$p = -2$
:$q = -3$
:$\map R x = e^{-x}$
First we establish the solution of the corresponding [[Definition:Constant Coefficient Homogeneous Li... | Linear Second Order ODE/y'' - 2 y' - 3 y = e^(-x) | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_3_y_=_e^(-x) | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_-_2_y'_-_3_y_=_e^(-x) | [
"Examples of Constant Coefficient LSOODEs",
"Examples of Method of Variation of Parameters"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Nonhomogeneous Linear Second Order ODE",
"Definition:Homogeneous Linear Second Order ODE with Constant Coefficients",
"Linear Second Order ODE/y'' - 2 y' - 3 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Primitive (Calculus)/Constant of Integration",
"Defini... |
proofwiki-23004 | Slice Category is Isomorphic to Identity Functor Over Object 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 $\mathbf C / C$ denote the slice category $\mathbf C$ over $C$.
Let $\paren {\operatorname{id}_{\mathbf C} \downarrow C}$ denote the comma category $\operatorname{id}_{... | Let $F: \mathbf 1\to \mathbf C$ be the functor defined by:
:$F* = C$
and
:$F \operatorname{id}_* = \operatorname{id}_C$
From Slice Category is Isomorphic to Comma Category:
:$\mathbf C / C \cong \paren {\operatorname{id}_{\mathbf C} \downarrow F}$
with isomorphisms of categories:
:$I_1: \mathbf C / C \to \paren {\opera... | 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 $\mathbf C / C$ denote the [[Definition:Slice Category|slice catego... | Let $F: \mathbf 1\to \mathbf C$ be the [[Definition:Covariant Functor|functor]] defined by:
:$F* = C$
and
:$F \operatorname{id}_* = \operatorname{id}_C$
From [[Slice Category is Isomorphic to Comma Category]]:
:$\mathbf C / C \cong \paren {\operatorname{id}_{\mathbf C} \downarrow F}$
with [[Definition:Isomorphism of... | Slice Category is Isomorphic to Identity Functor Over Object Comma Category | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Identity_Functor_Over_Object_Comma_Category | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Identity_Functor_Over_Object_Comma_Category | [
"Slice Categories",
"Comma Categories",
"Slice Category is Isomorphic to Identity Functor Over Object Comma Category"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Identity Functor",
"Definition:Slice Category",
"Definition:Comma Category/Functor Over Object",
"Definition:Isomorphism of Categories",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Commutative Dia... | [
"Definition:Functor/Covariant",
"Slice Category is Isomorphic to Comma Category",
"Definition:Isomorphism of Categories",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Functor Over Object Comma Category is Isomorphic to Com... |
proofwiki-23005 | Coslice Category is Isomorphic to Identity Functor Under Object 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 $C / \mathbf C$ denote the coslice category $\mathbf C$ under $C$.
Let $\paren {C \downarrow \operatorname{id}_{\mathbf C}}$ denote the comma category $\operatorname{id... | Let $G: \mathbf 1\to \mathbf C$ be the functor defined by:
:$G* = C$
and:
:$G \operatorname{id}_* = \operatorname{id}_C$
From Coslice Category is Isomorphic to Comma Category:
:$C / \mathbf C \cong \paren {G \downarrow \operatorname{id}_{\mathbf C} }$
with isomorphisms of categories:
:$I_1: C / \mathbf C \to \paren {G ... | 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 $C / \mathbf C$ denote the [[Definition:Coslice Category|coslice ca... | Let $G: \mathbf 1\to \mathbf C$ be the [[Definition:Covariant Functor|functor]] defined by:
:$G* = C$
and:
:$G \operatorname{id}_* = \operatorname{id}_C$
From [[Coslice Category is Isomorphic to Comma Category]]:
:$C / \mathbf C \cong \paren {G \downarrow \operatorname{id}_{\mathbf C} }$
with [[Definition:Isomorphis... | Coslice Category is Isomorphic to Identity Functor Under Object Comma Category | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Identity_Functor_Under_Object_Comma_Category | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Identity_Functor_Under_Object_Comma_Category | [
"Coslice Categories",
"Comma Categories",
"Coslice Category is Isomorphic to Identity Functor Under Object Comma Category"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Identity Functor",
"Definition:Coslice Category",
"Definition:Comma Category/Functor Under Object",
"Definition:Isomorphism of Categories",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Cate... | [
"Definition:Functor/Covariant",
"Coslice Category is Isomorphic to Comma Category",
"Definition:Isomorphism of Categories",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Functor Under Object Comma Category is Isomorphic to ... |
proofwiki-23006 | One-Sided Prepolar Reverses Subset Relation | Let $\innerprod E F$ be a dual system over $\R$.
Let $B, C \subseteq F$ be non-empty such that $B \subseteq C$.
Let $B_\odot$ and $C_\odot$ be the one-sided prepolars of $B$ and $C$ respectively.
Then:
:$C_\odot \subseteq B_\odot$ | Let $x \in C_\odot$.
Then we have:
:$\innerprod x f \le 1$ for all $f \in C$.
Since $f \in B$, we in particular have:
:$\innerprod x f \le 1$ for all $f \in B$.
Hence $x \in B_\odot$.
Hence $C_\odot \subseteq B_\odot$.
{{qed}}
Category:One-Sided Prepolars
kit9bf2tot7bcx9j6eguyz761ecamp0 | Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $B, C \subseteq F$ be [[Definition:Non-Empty Set|non-empty]] such that $B \subseteq C$.
Let $B_\odot$ and $C_\odot$ be the [[Definition:One-Sided Prepolar|one-sided prepolars]] of $B$ and $C$ respectively.
Then:
:$C_\odot \subseteq B_\... | Let $x \in C_\odot$.
Then we have:
:$\innerprod x f \le 1$ for all $f \in C$.
Since $f \in B$, we in particular have:
:$\innerprod x f \le 1$ for all $f \in B$.
Hence $x \in B_\odot$.
Hence $C_\odot \subseteq B_\odot$.
{{qed}}
[[Category:One-Sided Prepolars]]
kit9bf2tot7bcx9j6eguyz761ecamp0 | One-Sided Prepolar Reverses Subset Relation | https://proofwiki.org/wiki/One-Sided_Prepolar_Reverses_Subset_Relation | https://proofwiki.org/wiki/One-Sided_Prepolar_Reverses_Subset_Relation | [
"One-Sided Prepolars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Prepolar"
] | [
"Category:One-Sided Prepolars"
] |
proofwiki-23007 | One-Sided Prepolar of Dilation | Let $\innerprod E F$ be a dual system over $\R$.
Let $B \subseteq F$ be non-empty.
Let $\lambda > 0$.
Let $B_\odot$ and $\paren {\lambda B}_\odot$ be the one-sided polar of $B$ and $B_\odot`$ respectively.
Then:
:$\paren {\lambda B}_\odot = \dfrac 1 \lambda B_\odot$ | Let $x \in \dfrac 1 \lambda B_\odot$.
This is the case {{iff}} $\lambda x \in B_\odot$.
This is equivalent to $\innerprod {\lambda x} f \le 1$ for each $f \in B$.
Since $\innerprod \cdot \cdot$ is bilinear, we have $\innerprod {\lambda x} f = \innerprod x {\lambda f}$.
Hence $\innerprod x {\lambda f} \le 1$ for each $... | 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 $\lambda > 0$.
Let $B_\odot$ and $\paren {\lambda B}_\odot$ be the [[Definition:One-Sided Polar|one-sided polar]] of $B$ and $B_\odot`$ respectively.
Then:
:$\paren {\lamb... | Let $x \in \dfrac 1 \lambda B_\odot$.
This is the case {{iff}} $\lambda x \in B_\odot$.
This is equivalent to $\innerprod {\lambda x} f \le 1$ for each $f \in B$.
Since $\innerprod \cdot \cdot$ is [[Definition:Bilinear Mapping|bilinear]], we have $\innerprod {\lambda x} f = \innerprod x {\lambda f}$.
Hence $\inner... | One-Sided Prepolar of Dilation | https://proofwiki.org/wiki/One-Sided_Prepolar_of_Dilation | https://proofwiki.org/wiki/One-Sided_Prepolar_of_Dilation | [
"One-Sided Prepolars"
] | [
"Definition:Dual System",
"Definition:Non-Empty Set",
"Definition:One-Sided Polar"
] | [
"Definition:Bilinear Mapping",
"Category:One-Sided Prepolars"
] |
proofwiki-23008 | Linear Second Order ODE/y'' + k^2 y = 0/y(0) = 2, y'(0) = 1 | The second order ODE:
:$(1): \quad y' ' + k^2 y = 0$
with initial conditions:
:$\map y 0 = 2$
:$\map {y'} 0 = 1$
has the particular solution:
:$y = 2 \cos k x + \dfrac {\sin k x} k$ | From Linear Second Order ODE: $y' ' + k^2 y = 0$, the general solution of $(1)$ is:
:$y = C_1 \sin k x + C_2 \cos k x$
Differentiating {{WRT|Differentiation}} $x$:
:$y' = k C_1 \cos k x - k C_2 \sin k x$
Thus for the initial conditions:
{{begin-eqn}}
{{eqn | l = \map y 0
| r = C_1 \sin 0 + C_2 \cos 0
| c = ... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y' ' + k^2 y = 0$
with [[Definition:Initial Condition|initial conditions]]:
:$\map y 0 = 2$
:$\map {y'} 0 = 1$
has the [[Definition:Particular Solution of Differential Equation|particular solution]]:
:$y = 2 \cos k x + \dfrac {\sin k x} k$ | From [[Linear Second Order ODE/y'' + k^2 y = 0|Linear Second Order ODE: $y' ' + k^2 y = 0$]], the [[Definition:General Solution to Differential Equation|general solution]] of $(1)$ is:
:$y = C_1 \sin k x + C_2 \cos k x$
[[Definition:Differentiation|Differentiating]] {{WRT|Differentiation}} $x$:
:$y' = k C_1 \cos k x - ... | Linear Second Order ODE/y'' + k^2 y = 0/y(0) = 2, y'(0) = 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_k^2_y_=_0/y(0)_=_2,_y'(0)_=_1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_+_k^2_y_=_0/y(0)_=_2,_y'(0)_=_1 | [
"Linear Second Order ODE/y'' + k^2 y = 0"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution"
] | [
"Linear Second Order ODE/y'' + k^2 y = 0",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differentiation",
"Definition:Initial Condition"
] |
proofwiki-23009 | Proper Convex Combination of Interior of Convex Set and Closure of Convex Set is Contained in Interior of Set | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $C \subseteq X$ be a convex set with non-empty interior $\Int C$.
Let $\map \cl C$ be the closure of $C$ in $\struct {X, \tau}$.
Let $t \in \openint 0 1$.
Then we have:
:$t \Int C + \paren {1 - t} \map \cl C \subseteq \... | Let $x \in \Int C$ and $y \in \map \cl C$ so that $t x + \paren {1 - t} y \in t \Int C + \paren {1 - t} \map \cl C$ is general.
From Classification of Open Neighborhoods in Topological Vector Space, there exists an open neighborhood of ${\mathbf 0}_X$ in $\struct {X, \tau}$ such that $x + U \subseteq \Int C$.
From Dila... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $C \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex set]] with [[Definition:Non-Empty Set|non-empty]] [[Definition:Interior (Topology)|interior]] $\Int C$.
Let $\... | Let $x \in \Int C$ and $y \in \map \cl C$ so that $t x + \paren {1 - t} y \in t \Int C + \paren {1 - t} \map \cl C$ is general.
From [[Classification of Open Neighborhoods in Topological Vector Space]], there exists an [[Definition:Open Neighborhood|open neighborhood]] of ${\mathbf 0}_X$ in $\struct {X, \tau}$ such th... | Proper Convex Combination of Interior of Convex Set and Closure of Convex Set is Contained in Interior of Set | https://proofwiki.org/wiki/Proper_Convex_Combination_of_Interior_of_Convex_Set_and_Closure_of_Convex_Set_is_Contained_in_Interior_of_Set | https://proofwiki.org/wiki/Proper_Convex_Combination_of_Interior_of_Convex_Set_and_Closure_of_Convex_Set_is_Contained_in_Interior_of_Set | [
"Topological Vector Spaces",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Topological Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Non-Empty Set",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)"
] | [
"Classification of Open Neighborhoods in Topological Vector Space",
"Definition:Open Neighborhood",
"Dilation of Open Set in Topological Vector Space is Open",
"Definition:Open Neighborhood",
"Translation of Open Set in Topological Vector Space is Open",
"Definition:Open Neighborhood",
"Dilation of Open... |
proofwiki-23010 | Closure of Convex Set in Topological Vector Space with Non-Empty Interior coincides with Closure of Interior | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $C \subseteq X$ be a convex set with non-empty interior $\Int C$.
Let $\cl$ be the closure taken in $\struct {X, \tau}$.
Then:
:$\map \cl C = \map \cl {\Int C}$ | Firstly, from Topological Closure of Subset is Subset of Topological Closure, we have:
:$\map \cl {\Int C} \subseteq \map \cl C$
Conversely, take $x \in \map \cl C$.
Fix $z \in \Int C$.
From Proper Convex Combination of Interior of Convex Set and Closure of Convex Set is Contained in Interior of Set, we have:
:$\ds \f... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $C \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex set]] with [[Definition:Non-Empty Set|non-empty]] [[Definition:Interior (Topology)|interior]] $\Int C$.
Let $\... | Firstly, from [[Topological Closure of Subset is Subset of Topological Closure]], we have:
:$\map \cl {\Int C} \subseteq \map \cl C$
Conversely, take $x \in \map \cl C$.
Fix $z \in \Int C$.
From [[Proper Convex Combination of Interior of Convex Set and Closure of Convex Set is Contained in Interior of Set]], we hav... | Closure of Convex Set in Topological Vector Space with Non-Empty Interior coincides with Closure of Interior | https://proofwiki.org/wiki/Closure_of_Convex_Set_in_Topological_Vector_Space_with_Non-Empty_Interior_coincides_with_Closure_of_Interior | https://proofwiki.org/wiki/Closure_of_Convex_Set_in_Topological_Vector_Space_with_Non-Empty_Interior_coincides_with_Closure_of_Interior | [
"Topological Vector Spaces",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Topological Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Non-Empty Set",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)"
] | [
"Topological Closure of Subset is Subset of Topological Closure",
"Proper Convex Combination of Interior of Convex Set and Closure of Convex Set is Contained in Interior of Set",
"Multiple of Vector in Topological Vector Space Converges",
"Point in Set Closure iff Limit of Net"
] |
proofwiki-23011 | Open Neighborhood of Origin in Topological Vector Space is Absorbing | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $U$ be an open neighborhood of ${\mathbf 0}_X$ in $\struct {X, \tau}$.
Then $U$ is absorbing. | Define $m : \GF \times X \to X$ by:
:$\map m {\lambda, x} \mapsto \lambda x$
for any $\tuple {\lambda, x} \in \GF \times X$.
From the definition of a topological vector space, the mapping $f$ is continuous.
From Vertical Section of Continuous Function is Continuous, the mapping $m_x : \GF \to X$ defined by:
:$\map {m_x... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $U$ be an [[Definition:Open Neighborhood|open neighborhood]] of ${\mathbf 0}_X$ in $\struct {X, \tau}$.
Then $U$ is [[Definition:Absorbing Set|absorbing]]. | Define $m : \GF \times X \to X$ by:
:$\map m {\lambda, x} \mapsto \lambda x$
for any $\tuple {\lambda, x} \in \GF \times X$.
From the definition of a [[Definition:Topological Vector Space|topological vector space]], the [[Definition:Mapping|mapping]] $f$ is [[Definition:Continuous Mapping (Topology)|continuous]].
Fro... | Open Neighborhood of Origin in Topological Vector Space is Absorbing | https://proofwiki.org/wiki/Open_Neighborhood_of_Origin_in_Topological_Vector_Space_is_Absorbing | https://proofwiki.org/wiki/Open_Neighborhood_of_Origin_in_Topological_Vector_Space_is_Absorbing | [
"Topological Vector Spaces",
"Absorbing Sets"
] | [
"Definition:Topological Vector Space",
"Definition:Open Neighborhood",
"Definition:Absorbing Set"
] | [
"Definition:Topological Vector Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)",
"Vertical Section of Continuous Function is Continuous",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Absorbing Set",
"Category:Topological Vector Spaces",
"Categ... |
proofwiki-23012 | Interior of Closure of Convex Set in Topological Vector Space with Non-Empty Interior coincides with Interior of Set | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $C \subseteq X$ be a convex set with non-empty interior $\Int C$.
Let $\cl$ be the closure taken in $\struct {X, \tau}$.
Then we have:
:$\Int {\map \cl C} = \Int C$ | From Interior of Subset, we have $\Int C \subseteq \Int {\map \cl C}$.
We aim to show that $\Int {\map \cl C} \subseteq \Int C$.
Let $x \in \Int {\map \cl C}$.
Fix $x_0 \in \Int C$.
From Classification of Open Neighborhoods in Topological Vector Space, there exists an open neighborhood $W$ of ${\mathbf 0}_X$ in $\st... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $C \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex set]] with [[Definition:Non-Empty Set|non-empty]] [[Definition:Interior (Topology)|interior]] $\Int C$.
Let $\... | From [[Interior of Subset]], we have $\Int C \subseteq \Int {\map \cl C}$.
We aim to show that $\Int {\map \cl C} \subseteq \Int C$.
Let $x \in \Int {\map \cl C}$.
Fix $x_0 \in \Int C$.
From [[Classification of Open Neighborhoods in Topological Vector Space]], there exists an [[Definition:Open Neighborhood|open ... | Interior of Closure of Convex Set in Topological Vector Space with Non-Empty Interior coincides with Interior of Set | https://proofwiki.org/wiki/Interior_of_Closure_of_Convex_Set_in_Topological_Vector_Space_with_Non-Empty_Interior_coincides_with_Interior_of_Set | https://proofwiki.org/wiki/Interior_of_Closure_of_Convex_Set_in_Topological_Vector_Space_with_Non-Empty_Interior_coincides_with_Interior_of_Set | [
"Topological Vector Spaces",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Topological Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Non-Empty Set",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)"
] | [
"Interior of Subset",
"Classification of Open Neighborhoods in Topological Vector Space",
"Definition:Open Neighborhood",
"Open Neighborhood of Origin in Topological Vector Space is Absorbing",
"Definition:Absorbing Set",
"Proper Convex Combination of Interior of Convex Set and Closure of Convex Set is Co... |
proofwiki-23013 | Bound on Difference of Values of Convex Real Function | Let $X$ be a vector space over $\R$.
Let $C \subseteq X$ be a convex set.
Let $f : C \to \R$ be a convex function.
Let $x \in C$.
Let $z \in C$ be such that:
:$x + z \in C$ and $x - z \in C$.
Let $\delta \in \closedint 0 1$.
Then $x + \delta z \in C$, $x - \delta z \in C$, and further:
:$\cmod {\map f {x + \delta z} - ... | Write:
:$x + \delta z = \paren {1 - \delta} x + \delta \paren {x + z}$
and:
:$x - \delta z = \paren {1 - \delta} x + \delta \paren {x - z}$
Since $C$ is convex, we have $x + \delta z \in C$ and $x - \delta z \in C$.
Since $f$ is convex, we then have:
:$\map f {x + \delta z} \le \paren {1 - \delta} \map f x + \delta \m... | Let $X$ be a [[Definition:Vector Space|vector space]] 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/Vector Space|convex function]].
Let $x \in C$.
Let $z \in C$ be such that:
:$x + z \in C$ and $x - z \in C$.
Let $... | Write:
:$x + \delta z = \paren {1 - \delta} x + \delta \paren {x + z}$
and:
:$x - \delta z = \paren {1 - \delta} x + \delta \paren {x - z}$
Since $C$ is [[Definition:Convex Set (Vector Space)|convex]], we have $x + \delta z \in C$ and $x - \delta z \in C$.
Since $f$ is [[Definition:Convex Real Function/Vector Space|... | Bound on Difference of Values of Convex Real Function | https://proofwiki.org/wiki/Bound_on_Difference_of_Values_of_Convex_Real_Function | https://proofwiki.org/wiki/Bound_on_Difference_of_Values_of_Convex_Real_Function | [
"Convex Real Functions"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Real Function/Vector Space"
] | [
"Definition:Convex Set (Vector Space)",
"Definition:Convex Real Function/Vector Space",
"Definition:Convex Real Function/Vector Space"
] |
proofwiki-23014 | Local Criterion for Continuity of Convex Real Function | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $C \subseteq X$ be a convex set with non-empty interior $\Int C$.
Let $x \in \Int C$.
Let $U$ be an open neighborhood of $x$ in $\Int C$.
Let $f : C \to \R$ be a convex real function such that:
:$\ds \sup_{z \mathop \in ... | From Classification of Open Neighborhoods in Topological Vector Space, there exists an open neighborhood $V$ of ${\mathbf 0}_X$ such that:
:$U = x + V$
From Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods: Corollary 2, we can take $V$ to be balanced by passing to a subset.
In p... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $C \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex set]] with [[Definition:Non-Empty Set|non-empty]] [[Definition:Interior (Topology)|interior]] $\Int C$.
Let $x ... | From [[Classification of Open Neighborhoods in Topological Vector Space]], there exists an [[Definition:Open Neighborhood|open neighborhood]] $V$ of ${\mathbf 0}_X$ such that:
:$U = x + V$
From [[Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods/Corollary 2|Open Neighborhood of ... | Local Criterion for Continuity of Convex Real Function | https://proofwiki.org/wiki/Local_Criterion_for_Continuity_of_Convex_Real_Function | https://proofwiki.org/wiki/Local_Criterion_for_Continuity_of_Convex_Real_Function | [
"Topological Vector Spaces",
"Convex Real Functions"
] | [
"Definition:Topological Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Non-Empty Set",
"Definition:Interior (Topology)",
"Definition:Open Neighborhood",
"Definition:Convex Real Function/Vector Space",
"Definition:Continuous Mapping (Topology)/Point"
] | [
"Classification of Open Neighborhoods in Topological Vector Space",
"Definition:Open Neighborhood",
"Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods/Corollary 2",
"Definition:Balanced Set",
"Definition:Subset",
"Balanced Set in Vector Space is Symmetric",
"Defin... |
proofwiki-23015 | Lower Limit of Function is Bounded Above by Value at Limit Point | Let $\struct {X, \tau}$ be a topological space.
Let $f : X \to \overline \R$ be a function.
Let $\liminf$ be the lower limit.
Let $x_0 \in X$.
Then we have:
:$\ds \liminf_{x \mathop \to x_0} \map f x \le \map f {x_0}$ | Let $\map \mho {x_0}$ be the set of open neighborhoods of $x_0$ in $\struct {X, \tau}$.
Then for each $V \in \map \mho {x_0}$, we have:
:$\ds \inf_{x \mathop \in V} \map f x \le \map f {x_0}$
since $x_0 \in V$.
Hence:
:$\ds \sup_{V \in \map \mho {x_0} } \paren {\inf_{x \mathop \in V} \map f x} \le \map f {x_0}$
Hence... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f : X \to \overline \R$ be a [[Definition:Function|function]].
Let $\liminf$ be the [[Definition:Lower Limit|lower limit]].
Let $x_0 \in X$.
Then we have:
:$\ds \liminf_{x \mathop \to x_0} \map f x \le \map f {x_0}$ | Let $\map \mho {x_0}$ be the [[Definition:Set|set]] of [[Definition:Open Neighborhood|open neighborhoods]] of $x_0$ in $\struct {X, \tau}$.
Then for each $V \in \map \mho {x_0}$, we have:
:$\ds \inf_{x \mathop \in V} \map f x \le \map f {x_0}$
since $x_0 \in V$.
Hence:
:$\ds \sup_{V \in \map \mho {x_0} } \paren {\in... | Lower Limit of Function is Bounded Above by Value at Limit Point | https://proofwiki.org/wiki/Lower_Limit_of_Function_is_Bounded_Above_by_Value_at_Limit_Point | https://proofwiki.org/wiki/Lower_Limit_of_Function_is_Bounded_Above_by_Value_at_Limit_Point | [
"Lower Limit of Function is Bounded Above by Value at Limit Point",
"Limits of Mappings",
"Lower Limit of Function is Bounded Above by Value at Limit Point"
] | [
"Definition:Topological Space",
"Definition:Function",
"Definition:Lower Limit"
] | [
"Definition:Set",
"Definition:Open Neighborhood",
"Definition:Lower Limit",
"Category:Limits of Mappings",
"Category:Lower Limit of Function is Bounded Above by Value at Limit Point"
] |
proofwiki-23016 | Negative of Lower Limit is Upper Limit | Let $\struct {S, \tau}$ be a topological space.
Let $f : S \to \overline \R$ be an extended real valued function.
Let $x_0 \in S$.
Then:
:$\ds \limsup_{x \mathop \to x_0} \map f x = -\liminf_{x \mathop \to x_0} \paren {-\map f x}$
where $\limsup$ is the upper limit and $\liminf$ is the lower limit. | Let $\map \mho {x_0}$ be the set of open neighborhoods of $x_0$ in $\struct {X, \tau}$.
We then have:
{{begin-eqn}}
{{eqn | l = -\liminf_{x \mathop \to x_0} \paren {-\map f x}
| r = -\sup_{V \in \map \mho {x_0} } \paren {\inf_{x \mathop \in V} \paren {-\map f x} }
| c = {{Defof|Lower Limit (Topological Space)}}
}}
... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f : S \to \overline \R$ be an [[Definition:Extended Real-Valued Function|extended real valued function]].
Let $x_0 \in S$.
Then:
:$\ds \limsup_{x \mathop \to x_0} \map f x = -\liminf_{x \mathop \to x_0} \paren {-\map f x}$
where $... | Let $\map \mho {x_0}$ be the [[Definition:Set|set]] of [[Definition:Open Neighborhood|open neighborhoods]] of $x_0$ in $\struct {X, \tau}$.
We then have:
{{begin-eqn}}
{{eqn | l = -\liminf_{x \mathop \to x_0} \paren {-\map f x}
| r = -\sup_{V \in \map \mho {x_0} } \paren {\inf_{x \mathop \in V} \paren {-\map f x} }
... | Negative of Lower Limit is Upper Limit | https://proofwiki.org/wiki/Negative_of_Lower_Limit_is_Upper_Limit | https://proofwiki.org/wiki/Negative_of_Lower_Limit_is_Upper_Limit | [
"Lower Limits",
"Lower Limits (Topological Spaces)",
"Upper Limits",
"Upper Limits (Topological Spaces)",
"Lower Limits (Topological Spaces)",
"Upper Limits (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Extended Real-Valued Function",
"Definition:Upper Limit (Topological Space)",
"Definition:Lower Limit (Topological Space)"
] | [
"Definition:Set",
"Definition:Open Neighborhood",
"Negative of Supremum is Infimum of Negatives",
"Negative of Infimum is Supremum of Negatives",
"Category:Lower Limits (Topological Spaces)",
"Category:Upper Limits (Topological Spaces)"
] |
proofwiki-23017 | Function is Upper Semicontinuous iff Negative is Lower Semicontinuous | Let $\struct {S, \tau}$ be a topological space.
Let $f : S \to \overline \R$ be an extended real valued function.
Let $x_0 \in S$.
Then $f$ is upper semicontinuous at $x_0$ {{iff}} $-f$ is lower semicontinuous at $x_0$. | We have that $f$ is upper semicontinuous at $x_0$ {{iff}}:
:$\ds \map f {x_0} = \limsup_{x \mathop \to x_0} \map f x$
where $\limsup$ is the upper limit.
This is equivalent to:
:$\ds -\map f {x_0} = -\limsup_{x \mathop \to x_0} \map f x$
From Negative of Lower Limit is Upper Limit, we have:
:$\ds -\limsup_{x \mathop \t... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f : S \to \overline \R$ be an [[Definition:Extended Real-Valued Function|extended real valued function]].
Let $x_0 \in S$.
Then $f$ is [[Definition:Upper Semicontinuous|upper semicontinuous]] at $x_0$ {{iff}} $-f$ is [[Definition:... | We have that $f$ is [[Definition:Upper Semicontinuous|upper semicontinuous]] at $x_0$ {{iff}}:
:$\ds \map f {x_0} = \limsup_{x \mathop \to x_0} \map f x$
where $\limsup$ is the [[Definition:Upper Limit (Topological Space)|upper limit]].
This is equivalent to:
:$\ds -\map f {x_0} = -\limsup_{x \mathop \to x_0} \map f x... | Function is Upper Semicontinuous iff Negative is Lower Semicontinuous | https://proofwiki.org/wiki/Function_is_Upper_Semicontinuous_iff_Negative_is_Lower_Semicontinuous | https://proofwiki.org/wiki/Function_is_Upper_Semicontinuous_iff_Negative_is_Lower_Semicontinuous | [
"Upper Semicontinuity",
"Lower Semicontinuity"
] | [
"Definition:Topological Space",
"Definition:Extended Real-Valued Function",
"Definition:Upper Semicontinuous",
"Definition:Lower Semicontinuous"
] | [
"Definition:Upper Semicontinuous",
"Definition:Upper Limit (Topological Space)",
"Negative of Lower Limit is Upper Limit",
"Definition:Lower Limit (Topological Space)",
"Definition:Upper Semicontinuous",
"Definition:Lower Semicontinuous",
"Category:Upper Semicontinuity",
"Category:Lower Semicontinuity... |
proofwiki-23018 | Characterization of Upper Semicontinuity | Let $\struct {S, \tau}$ be a topological space.
Let $f: S \to \overline \R$ be an extended real valued function.
{{TFAE}}
:$(1): \quad$ $f$ is upper semicontinuous
:$(2): \quad$ $\set {x \in S : \map f x \ge \alpha}$ is closed for each $\alpha \in \R$
:$(3): \quad$ the hypograph $\map {\operatorname {hypo} } f$ is clos... | === $(1)$ implies $(2)$ ===
Suppose that $f$ is upper semicontinuous.
From Function is Upper Semicontinuous iff Negative is Lower Semicontinuous, $-f$ is lower semicontinuous.
From Characterization of Lower Semicontinuity, we have that:
:$\set {x \in S : -\map f x \le -\alpha}$ is closed for each $\alpha \in \R$.
Hence... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f: S \to \overline \R$ be an [[Definition:Extended Real-Valued Function|extended real valued function]].
{{TFAE}}
:$(1): \quad$ $f$ is [[Definition:Upper Semicontinuous|upper semicontinuous]]
:$(2): \quad$ $\set {x \in S : \map f x... | === $(1)$ implies $(2)$ ===
Suppose that $f$ is [[Definition:Upper Semicontinuous|upper semicontinuous]].
From [[Function is Upper Semicontinuous iff Negative is Lower Semicontinuous]], $-f$ is [[Definition:Lower Semicontinuous|lower semicontinuous]].
From [[Characterization of Lower Semicontinuity]], we have that:
... | Characterization of Upper Semicontinuity | https://proofwiki.org/wiki/Characterization_of_Upper_Semicontinuity | https://proofwiki.org/wiki/Characterization_of_Upper_Semicontinuity | [
"Upper Semicontinuity"
] | [
"Definition:Topological Space",
"Definition:Extended Real-Valued Function",
"Definition:Upper Semicontinuous",
"Definition:Closed Set",
"Definition:Hypograph",
"Definition:Closed Set"
] | [
"Definition:Upper Semicontinuous",
"Function is Upper Semicontinuous iff Negative is Lower Semicontinuous",
"Definition:Lower Semicontinuous",
"Characterization of Lower Semicontinuity",
"Definition:Closed Set",
"Definition:Closed Set",
"Definition:Closed Set",
"Definition:Closed Set",
"Definition:C... |
proofwiki-23019 | Global Criterion for Continuity of Convex Real Function | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space over $\GF$.
Let $C \subseteq X$ be an open convex set.
Let $f : C \to \R$ be a convex function.
{{TFAE}}
:$(1): \quad$ $f$ is continuous
:$(2): \quad$ $f$ is upper semicontinuous
:$(3): \quad$ for each $x \in C$, there exists an open nei... | === $(1)$ implies $(2)$ ===
This follows from Continuity implies Upper Semicontinuity.
{{qed|lemma}} | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $C \subseteq X$ be an [[Definition:Open Set|open]] [[Definition:Convex Set (Vector Space)|convex set]].
Let $f : C \to \R$ be a [[Definition:Convex Real Function/Vector Space|con... | === $(1)$ implies $(2)$ ===
This follows from [[Continuity implies Upper Semicontinuity]].
{{qed|lemma}} | Global Criterion for Continuity of Convex Real Function | https://proofwiki.org/wiki/Global_Criterion_for_Continuity_of_Convex_Real_Function | https://proofwiki.org/wiki/Global_Criterion_for_Continuity_of_Convex_Real_Function | [
"Convex Real Functions"
] | [
"Definition:Topological Vector Space",
"Definition:Open Set",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Real Function/Vector Space",
"Definition:Continuous Mapping",
"Definition:Upper Semicontinuous",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Contin... | [
"Continuity implies Upper Semicontinuity"
] |
proofwiki-23020 | Continuity implies Upper Semicontinuity | Let $\struct {S, \tau}$ be a topological space.
Let $f : S \to \R$ be a continuous function.
Then $f$ is upper semicontinuous. | Let $x_0 \in S$.
Let $\map \mho {x_0}$ be the set of open neighborhoods of $x_0$.
Let $\epsilon > 0$.
Since $f$ is continuous, it is continuous at $x_0$.
From the definition of continuity at $x_0$, there exists an open neighborhood $U$ of $x_0$ such that:
:$f \sqbrk U \subseteq \openint {\map f {x_0} - \epsilon} {\map... | Let $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f : S \to \R$ be a [[Definition:Continuous Mapping|continuous function]].
Then $f$ is [[Definition:Upper Semicontinuous|upper semicontinuous]]. | Let $x_0 \in S$.
Let $\map \mho {x_0}$ be the [[Definition:Set|set]] of [[Definition:Open Neighborhood|open neighborhoods]] of $x_0$.
Let $\epsilon > 0$.
Since $f$ is [[Definition:Continuous Mapping|continuous]], it is [[Definition:Continuous Mapping at Point (Topology)|continuous at $x_0$]].
From the definition o... | Continuity implies Upper Semicontinuity | https://proofwiki.org/wiki/Continuity_implies_Upper_Semicontinuity | https://proofwiki.org/wiki/Continuity_implies_Upper_Semicontinuity | [
"Upper Semicontinuity"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping",
"Definition:Upper Semicontinuous"
] | [
"Definition:Set",
"Definition:Open Neighborhood",
"Definition:Continuous Mapping",
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Open Neighborhood",
"Upper Limit of Function is Bounded Below by Value at Limit Point/Corollary",
"Definiti... |
proofwiki-23021 | Upper Limit of Function is Bounded Below by Value at Limit Point | Let $\struct {X, \tau}$ be a topological space.
Let $f : X \to \overline \R$ be a function.
Let $\limsup$ be the upper limit.
Let $x_0 \in X$.
Then we have:
:$\ds \map f {x_0} \le \limsup_{x \mathop \to x_0} \map f x$ | Let $\map \mho {x_0}$ be the set of open neighborhoods of $x_0$ in $\struct {X, \tau}$.
Then for each $V \in \map \mho {x_0}$, we have:
:$\ds \sup_{x \mathop \in V} \map f x \ge \map f {x_0}$
since $x_0 \in V$.
Hence:
:$\ds \inf_{V \in \map \mho {x_0} } \paren {\sup_{x \mathop \in V} \map f x} \ge \map f {x_0}$
Hence... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f : X \to \overline \R$ be a [[Definition:Function|function]].
Let $\limsup$ be the [[Definition:Upper Limit (Topological Space)|upper limit]].
Let $x_0 \in X$.
Then we have:
:$\ds \map f {x_0} \le \limsup_{x \mathop \to x_0} \ma... | Let $\map \mho {x_0}$ be the [[Definition:Set|set]] of [[Definition:Open Neighborhood|open neighborhoods]] of $x_0$ in $\struct {X, \tau}$.
Then for each $V \in \map \mho {x_0}$, we have:
:$\ds \sup_{x \mathop \in V} \map f x \ge \map f {x_0}$
since $x_0 \in V$.
Hence:
:$\ds \inf_{V \in \map \mho {x_0} } \paren {\su... | Upper Limit of Function is Bounded Below by Value at Limit Point | https://proofwiki.org/wiki/Upper_Limit_of_Function_is_Bounded_Below_by_Value_at_Limit_Point | https://proofwiki.org/wiki/Upper_Limit_of_Function_is_Bounded_Below_by_Value_at_Limit_Point | [
"Limits of Mappings",
"Upper Limit of Function is Bounded Below by Value at Limit Point"
] | [
"Definition:Topological Space",
"Definition:Function",
"Definition:Upper Limit (Topological Space)"
] | [
"Definition:Set",
"Definition:Open Neighborhood",
"Definition:Upper Limit (Topological Space)",
"Category:Limits of Mappings",
"Category:Upper Limit of Function is Bounded Below by Value at Limit Point"
] |
proofwiki-23022 | Continuity of Convex Real Function at Interior Point implies Local Lipschitz Continuity | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $C \subseteq X$ be an open convex set with non-empty interior $\Int C$.
Let $x \in \Int C$.
Let $f : C \to \R$ be a convex function such that:
:$f$ is continuous at $x$.
Let $\map {B_\delta} x$ denote the op... | Since $f$ is continuous at $x$, there exists $\delta > 0$ such that:
:$\norm {u - x}_X < 2 \delta$
implies that:
:$\size {\map f u - \map f x} < \dfrac 1 2$
for $u \in C$.
Then if:
:$\norm {w - x}_X < 2 \delta$
and:
:$\norm {z - x}_X < 2 \delta$
we have:
:$\size {\map f w - \map f z} < 1$
by the Triangle Inequality.
T... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $C \subseteq X$ be an [[Definition:Open Set|open]] [[Definition:Convex Set (Vector Space)|convex set]] with [[Definition:Non-Empty Set|non-empty]] [[Definition:Interior (To... | Since $f$ is [[Definition:Continuous Mapping|continuous]] at $x$, there exists $\delta > 0$ such that:
:$\norm {u - x}_X < 2 \delta$
implies that:
:$\size {\map f u - \map f x} < \dfrac 1 2$
for $u \in C$.
Then if:
:$\norm {w - x}_X < 2 \delta$
and:
:$\norm {z - x}_X < 2 \delta$
we have:
:$\size {\map f w - \map f z}... | Continuity of Convex Real Function at Interior Point implies Local Lipschitz Continuity | https://proofwiki.org/wiki/Continuity_of_Convex_Real_Function_at_Interior_Point_implies_Local_Lipschitz_Continuity | https://proofwiki.org/wiki/Continuity_of_Convex_Real_Function_at_Interior_Point_implies_Local_Lipschitz_Continuity | [
"Lipschitz Continuity",
"Convex Real Functions"
] | [
"Definition:Normed Vector Space",
"Definition:Open Set",
"Definition:Convex Set (Vector Space)",
"Definition:Non-Empty Set",
"Definition:Interior (Topology)",
"Definition:Convex Real Function/Vector Space",
"Definition:Continuous Mapping",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"De... | [
"Definition:Continuous Mapping",
"Triangle Inequality",
"Definition:Convex Real Function/Vector Space",
"Definition:Lipschitz Continuity",
"Definition:Lipschitz Continuity/Lipschitz Constant"
] |
proofwiki-23023 | Separation of Non-Empty Set from Set containing Interior Point is Continuous and Proper | Let $\struct {X, \tau}$ be a topological vector space over $\R$.
Let $B \subseteq X$ be a non-empty set.
Let $A \subseteq X$ have non-empty interior $\Int A$ such that there exists a linear functional $f : X \to \R$ and $\alpha \in \R$ with:
:$\map f b \le \alpha \le \map f a$ for each $a \in A$ and $b \in B$.
That is... | By Non-Zero Linear Functional on Topological Vector Space does not attain Minimum on Interior of Set, we have that $f$ is continuous.
Further, Non-Zero Linear Functional on Topological Vector Space does not attain Minimum on Interior of Set gives that:
:$\map f a > \alpha$ for each $a \in \Int A$.
Since $\Int A \ne \O$... | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\R$.
Let $B \subseteq X$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $A \subseteq X$ have [[Definition:Non-Empty Set|non-empty]] [[Definition:Interior (Topology)|interior]] $\Int A$ such that there exists a [[... | By [[Non-Zero Linear Functional on Topological Vector Space does not attain Minimum on Interior of Set]], we have that $f$ is [[Definition:Continuous Mapping|continuous]].
Further, [[Non-Zero Linear Functional on Topological Vector Space does not attain Minimum on Interior of Set]] gives that:
:$\map f a > \alpha$ for... | Separation of Non-Empty Set from Set containing Interior Point is Continuous and Proper | https://proofwiki.org/wiki/Separation_of_Non-Empty_Set_from_Set_containing_Interior_Point_is_Continuous_and_Proper | https://proofwiki.org/wiki/Separation_of_Non-Empty_Set_from_Set_containing_Interior_Point_is_Continuous_and_Proper | [
"Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Non-Empty Set",
"Definition:Non-Empty Set",
"Definition:Interior (Topology)",
"Definition:Linear Functional",
"Definition:Affine Hyperplane",
"Definition:Affine Hyperplane Separating Two Sets",
"Definition:Continuous Mapping",
"Definition:Affine Hyp... | [
"Non-Zero Linear Functional on Topological Vector Space does not attain Minimum on Interior of Set",
"Definition:Continuous Mapping",
"Non-Zero Linear Functional on Topological Vector Space does not attain Minimum on Interior of Set",
"Definition:Affine Hyperplane Separating Two Sets",
"Definition:Affine Hy... |
proofwiki-23024 | Interior Separating Hyperplane Theorem | Let $\struct {X, \tau}$ be a topological vector space over $\R$.
Let $\cl$ be the closure in $\struct {X, \tau}$.
Let $A \subseteq X$ be a convex set with non-empty interior $\Int A$.
Let $B \subseteq X$ be a non-empty convex set with $A \cap B = \O$.
Then $\map \cl A$ and $\map \cl B$ are properly separated by a co... | Since $A \cap B = \O$, we have $\Int A \cap B = \O$.
From Interior of Convex Set in Topological Vector Space is Convex, $\Int A$ is convex.
From Interior Point of Subset of Topological Vector Space is Internal Point, each point in $\Int A$ is an internal point of $A$.
Hence, from Basic Separating Hyperplane Theorem, t... | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\R$.
Let $\cl$ be the [[Definition:Closure (Topology)|closure]] in $\struct {X, \tau}$.
Let $A \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex set]] with [[Definition:Non-Empty Set|non-empty]] [[Defini... | Since $A \cap B = \O$, we have $\Int A \cap B = \O$.
From [[Interior of Convex Set in Topological Vector Space is Convex]], $\Int A$ is [[Definition:Convex Set (Vector Space)|convex]].
From [[Interior Point of Subset of Topological Vector Space is Internal Point]], each point in $\Int A$ is an [[Definition:Internal P... | Interior Separating Hyperplane Theorem | https://proofwiki.org/wiki/Interior_Separating_Hyperplane_Theorem | https://proofwiki.org/wiki/Interior_Separating_Hyperplane_Theorem | [
"Affine Hyperplanes"
] | [
"Definition:Topological Vector Space",
"Definition:Closure (Topology)",
"Definition:Convex Set (Vector Space)",
"Definition:Non-Empty Set",
"Definition:Interior (Topology)",
"Definition:Non-Empty Set",
"Definition:Convex Set (Vector Space)",
"Definition:Affine Hyperplane Separating Two Sets/Proper Sep... | [
"Interior of Convex Set in Topological Vector Space is Convex",
"Definition:Convex Set (Vector Space)",
"Interior Point of Subset of Topological Vector Space is Internal Point",
"Definition:Internal Point of Subset of Vector Space",
"Basic Separating Hyperplane Theorem",
"Definition:Linear Functional",
... |
proofwiki-23025 | Fan-Glicksberg-Hoffman Concave Alternative | Let $X$ be a vector space over $\R$.
Let $C \subseteq X$ be a convex set.
Let $f_1, \ldots, f_m : C \to \R$ be concave functions.
Then exactly one of the following is true:
:$(1): \quad$ there exists $x \in C$ such that $\map {f_i} x > 0$ for each $1 \le i \le n$
:$(2): \quad$ there exists $\lambda_1, \ldots, \lambda_n... | We show that $(1)$ and $(2)$ cannot hold simultaneously.
Suppose that $(1)$ holds.
Then:
:there exists $x \in C$ such that $\map {f_i} x > 0$ for each $1 \le i \le n$.
Let $\lambda_1, \ldots, \lambda_n \in \R_{\ge 0}$ be such that $\lambda_j \ne 0$.
Then we have:
:$\ds \sum_{i \mathop = 1}^n \lambda_i \map {f_i} x \ge... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\R$.
Let $C \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex set]].
Let $f_1, \ldots, f_m : C \to \R$ be [[Definition:Concave Real Function/Vector Space|concave functions]].
Then exactly one of the following is true:
:$(1): \quad$ there exis... | We show that $(1)$ and $(2)$ cannot hold simultaneously.
Suppose that $(1)$ holds.
Then:
:there exists $x \in C$ such that $\map {f_i} x > 0$ for each $1 \le i \le n$.
Let $\lambda_1, \ldots, \lambda_n \in \R_{\ge 0}$ be such that $\lambda_j \ne 0$.
Then we have:
:$\ds \sum_{i \mathop = 1}^n \lambda_i \map {f_i} x... | Fan-Glicksberg-Hoffman Concave Alternative | https://proofwiki.org/wiki/Fan-Glicksberg-Hoffman_Concave_Alternative | https://proofwiki.org/wiki/Fan-Glicksberg-Hoffman_Concave_Alternative | [
"Concave Real Functions"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Concave Real Function/Vector Space"
] | [
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Set of Vectors in Euclidean Space with all Components Non-Negative is Convex",
"Definition:Convex Set (Vector Space)",
"Interior of Set of Vectors in Euclidean Space with all Components... |
proofwiki-23026 | Interior Point of Subset of Topological Vector Space is Internal Point | Let $\struct {X, \tau}$ be a topological vector space over $\R$.
Let $A \subseteq X$ be a non-empty.
Let $x \in \Int A$.
Then $x$ is an internal point of $A$. | Since $x \in \Int A$, there exists an open neighborhood $U$ of $x$ such that:
:$U \subseteq \Int A$
From Classification of Open Neighborhoods in Topological Vector Space, there exists an open neighborhood $V$ of ${\mathbf 0}_X$ such that:
:$U = x + V$
Then:
:$x + V \subseteq A$
From Open Neighborhood of Origin in Topol... | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\R$.
Let $A \subseteq X$ be a [[Definition:Non-Empty Set|non-empty]].
Let $x \in \Int A$.
Then $x$ is an [[Definition:Internal Point of Subset of Vector Space|internal point]] of $A$. | Since $x \in \Int A$, there exists an [[Definition:Open Neighborhood|open neighborhood]] $U$ of $x$ such that:
:$U \subseteq \Int A$
From [[Classification of Open Neighborhoods in Topological Vector Space]], there exists an [[Definition:Open Neighborhood|open neighborhood]] $V$ of ${\mathbf 0}_X$ such that:
:$U = x + ... | Interior Point of Subset of Topological Vector Space is Internal Point | https://proofwiki.org/wiki/Interior_Point_of_Subset_of_Topological_Vector_Space_is_Internal_Point | https://proofwiki.org/wiki/Interior_Point_of_Subset_of_Topological_Vector_Space_is_Internal_Point | [
"Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Non-Empty Set",
"Definition:Internal Point of Subset of Vector Space"
] | [
"Definition:Open Neighborhood",
"Classification of Open Neighborhoods in Topological Vector Space",
"Definition:Open Neighborhood",
"Open Neighborhood of Origin in Topological Vector Space is Absorbing",
"Definition:Absorbing Set",
"Definition:Internal Point of Subset of Vector Space",
"Category:Topolog... |
proofwiki-23027 | Sublinear Functional is Linear Functional iff Dominates Unique Linear Functional | Let $X$ be a vector space over $\R$.
Let $p : X \to \R$ be a sublinear functional.
Then $p$ is linear {{iff}} there exists a unique linear functional $f : X \to \R$ such that:
:$\map f x \le \map p x$ for each $x \in X$. | === Necessary Condition ===
Suppose that $p$ is linear.
Let $f : X \to \R$ be a linear functional such that:
:$\map f x \le \map p x$ for each $x \in X$.
Then, swapping $x$ with $-x$ we have:
:$\map f {-x} \le \map p {-x}$ for each $x \in X$.
That is, since both $f$ and $p$ are linear, we have:
:$-\map f x \le -\map p ... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\R$.
Let $p : X \to \R$ be a [[Definition:Sublinear Functional|sublinear functional]].
Then $p$ is [[Definition:Linear Functional|linear]] {{iff}} there exists a unique [[Definition:Linear Functional|linear functional]] $f : X \to \R$ such that:
:$\map f x... | === Necessary Condition ===
Suppose that $p$ is [[Definition:Linear Functional|linear]].
Let $f : X \to \R$ be a [[Definition:Linear Functional|linear functional]] such that:
:$\map f x \le \map p x$ for each $x \in X$.
Then, swapping $x$ with $-x$ we have:
:$\map f {-x} \le \map p {-x}$ for each $x \in X$.
That is... | Sublinear Functional is Linear Functional iff Dominates Unique Linear Functional | https://proofwiki.org/wiki/Sublinear_Functional_is_Linear_Functional_iff_Dominates_Unique_Linear_Functional | https://proofwiki.org/wiki/Sublinear_Functional_is_Linear_Functional_iff_Dominates_Unique_Linear_Functional | [
"Sublinear Functionals"
] | [
"Definition:Vector Space",
"Definition:Sublinear Functional",
"Definition:Linear Functional",
"Definition:Linear Functional"
] | [
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definitio... |
proofwiki-23028 | Convex Hahn-Banach Theorem | Let $X$ be a vector space over $\R$.
Let $M$ be a vector subspace of $X$.
Let $p : X \to \R$ be a convex function.
Let $f : M \to \R$ be a linear functional satisfying:
:$\map f x \le \map p x$ for each $x \in M$.
Then there exists a linear functional $\hat f : X \to \R$ extending $f$ and satisfying:
:$\map {\hat f} x... | Define $P : X \to \hointr 0 \infty$ by:
:$\ds \map P x = \inf_{t > 0} t^{-1} \map p {t x}$ for each $x \in X$.
Since $\map p x \ge 0$ for each $x \in X$, we have that $P$ is well-defined.
We can also see, by taking $t = 1$, that:
:$\map P x \le \map p x$ for each $x \in X$.
We show that $P$ is a sublinear functional.
L... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\R$.
Let $M$ be a [[Definition:Vector Subspace|vector subspace]] of $X$.
Let $p : X \to \R$ be a [[Definition:Convex Real Function/Vector Space|convex function]].
Let $f : M \to \R$ be a [[Definition:Linear Functional|linear functional]] satisfying:
:$\map... | Define $P : X \to \hointr 0 \infty$ by:
:$\ds \map P x = \inf_{t > 0} t^{-1} \map p {t x}$ for each $x \in X$.
Since $\map p x \ge 0$ for each $x \in X$, we have that $P$ is well-defined.
We can also see, by taking $t = 1$, that:
:$\map P x \le \map p x$ for each $x \in X$.
We show that $P$ is a [[Definition:Sublin... | Convex Hahn-Banach Theorem | https://proofwiki.org/wiki/Convex_Hahn-Banach_Theorem | https://proofwiki.org/wiki/Convex_Hahn-Banach_Theorem | [
"Hahn-Banach Theorem",
"Convex Real Functions"
] | [
"Definition:Vector Space",
"Definition:Vector Subspace",
"Definition:Convex Real Function/Vector Space",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Extension of Mapping"
] | [
"Definition:Sublinear Functional",
"Multiple of Infimum",
"Definition:Homogeneous Function/Positive Homogeneity",
"Definition:Infimum of Mapping",
"Definition:Infimum of Mapping",
"Definition:Sublinear Functional",
"Definition:Linear Functional",
"Definition:Infimum of Mapping",
"Hahn-Banach Theorem... |
proofwiki-23029 | Normed Vector Space is Finite Dimensional iff Closed Unit Ball is Weak Neighborhood of Origin | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ is a normed vector space over $\GF$.
Let $B_X^-$ be the closed unit ball in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w$ be the weak topology of $\struct {X, \norm {\, \cdot \,}_X}$.
Then $X$ is finite-dimensional {{iff}} $B_X^-$ is a $w$-neighborho... | From the definition of the weak topology, every $w$-open set is norm open.
We show that under the hypotheses, every norm open set is $w$-open.
Let $U$ be a norm open set.
Let $V \subseteq B_X^-$ be $w$-open.
Then for each $x \in U$, there exists $\delta_x > 0$ such that:
:$x + \delta_x B_X^- \subseteq U$
so that:
:$x +... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ is a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $B_X^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w$ be the [[Definition:Weak Topology|weak topology]] of $\str... | From the definition of the [[Definition:Weak Topology|weak topology]], every [[Definition:Weakly Open Set|$w$-open set]] is [[Definition:Open Set/Normed Vector Space|norm open]].
We show that under the hypotheses, every [[Definition:Open Set/Normed Vector Space|norm open set]] is [[Definition:Weakly Open Set|$w$-open]... | Normed Vector Space is Finite Dimensional iff Closed Unit Ball is Weak Neighborhood of Origin | https://proofwiki.org/wiki/Normed_Vector_Space_is_Finite_Dimensional_iff_Closed_Unit_Ball_is_Weak_Neighborhood_of_Origin | https://proofwiki.org/wiki/Normed_Vector_Space_is_Finite_Dimensional_iff_Closed_Unit_Ball_is_Weak_Neighborhood_of_Origin | [
"Weak Topologies",
"Weak Topologies on Topological Vector Spaces",
"Normed Vector Spaces",
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Unit Ball",
"Definition:Initial Topology",
"Definition:Dimension of Vector Space/Finite",
"Definition:Neighborhood"
] | [
"Definition:Initial Topology",
"Definition:Weakly Open Set",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Weakly Open Set",
"Definition:Open Set/Normed Vector Space",
"Definition:Weakly Open Set",
"Dilation of Open Set in Topological Vector Space is... |
proofwiki-23030 | Closed Unit Ball of Infinite-Dimensional Normed Vector Space has Empty Weak Interior | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$ that is infinite-dimensional.
Let $B_X^-$ be the closed unit ball of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\operatorname {Int}_w$ be the in... | {{AimForCont}} that $\map {\operatorname {Int}_w} {B_X^-} \ne \O$.
Let $u \in \map {\operatorname {Int}_w} {B_X^-}$.
Then there exists an open neighborhood $U$ of $u$ such that $U \subseteq B_X^-$.
From Classification of Open Neighborhoods in Topological Vector Space, there exists an open neighborhood $V$ of ${\mathbf ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$ that is [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]].
Let $B_X^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] of $\struct {X, \norm {\, \cdo... | {{AimForCont}} that $\map {\operatorname {Int}_w} {B_X^-} \ne \O$.
Let $u \in \map {\operatorname {Int}_w} {B_X^-}$.
Then there exists an [[Definition:Open Neighborhood|open neighborhood]] $U$ of $u$ such that $U \subseteq B_X^-$.
From [[Classification of Open Neighborhoods in Topological Vector Space]], there exist... | Closed Unit Ball of Infinite-Dimensional Normed Vector Space has Empty Weak Interior | https://proofwiki.org/wiki/Closed_Unit_Ball_of_Infinite-Dimensional_Normed_Vector_Space_has_Empty_Weak_Interior | https://proofwiki.org/wiki/Closed_Unit_Ball_of_Infinite-Dimensional_Normed_Vector_Space_has_Empty_Weak_Interior | [
"Weak Topologies on Topological Vector Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Infinite-Dimensional Vector Space",
"Definition:Closed Unit Ball",
"Definition:Initial Topology",
"Definition:Interior (Topology)"
] | [
"Definition:Open Neighborhood",
"Classification of Open Neighborhoods in Topological Vector Space",
"Definition:Open Neighborhood",
"Dilation of Open Set in Topological Vector Space is Open",
"Definition:Weakly Open Set",
"Definition:Open Neighborhood",
"Definition:Neighborhood",
"Normed Vector Space ... |
proofwiki-23031 | Intersection of Finitely Many Kernels of Linear Functionals in Infinite-Dimensional Vector Space is Infinite-Dimensional with Finite Codimension | Let $K$ be a field.
Let $X$ be an infinite-dimensional vector space over $K$.
Let $X^\#$ be the algebraic dual of $X$.
Let $F$ be a non-empty finite subset of $X^\#$.
We then have that:
:$\ds \map {\operatorname {dim} } {\bigcap_{f \in F} \map \ker f} = \infty$
and:
:$\ds \map {\operatorname {codim} } {\bigcap_{F \in ... | Let $\operatorname {codim}$ denote codimension.
From Kernel of Linear Transformation is Linear Subspace, $\ker f$ is a vector subspace for each $f \in F$.
From Set of Linear Subspaces is Closed under Intersection:
:$\ds \bigcap_{f \in F} \ker f$ is a vector subspace of $X$.
If $f = {\mathbf 0}_{X^\#}$, then $\map \ker ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Vector Space|vector space]] over $K$.
Let $X^\#$ be the [[Definition:Algebraic Dual|algebraic dual]] of $X$.
Let $F$ be a [[Definition:Non-Empty Set|non-empty]]... | Let $\operatorname {codim}$ denote [[Definition:Codimension of Vector Subspace|codimension]].
From [[Kernel of Linear Transformation is Linear Subspace]], $\ker f$ is a [[Definition:Vector Subspace|vector subspace]] for each $f \in F$.
From [[Set of Linear Subspaces is Closed under Intersection]]:
:$\ds \bigcap_{f \i... | Intersection of Finitely Many Kernels of Linear Functionals in Infinite-Dimensional Vector Space is Infinite-Dimensional with Finite Codimension | https://proofwiki.org/wiki/Intersection_of_Finitely_Many_Kernels_of_Linear_Functionals_in_Infinite-Dimensional_Vector_Space_is_Infinite-Dimensional_with_Finite_Codimension | https://proofwiki.org/wiki/Intersection_of_Finitely_Many_Kernels_of_Linear_Functionals_in_Infinite-Dimensional_Vector_Space_is_Infinite-Dimensional_with_Finite_Codimension | [
"Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Infinite-Dimensional Vector Space",
"Definition:Vector Space",
"Definition:Algebraic Dual",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Kernel of Linear Functional"
] | [
"Definition:Codimension of Vector Subspace",
"Kernel of Linear Transformation is Linear Subspace",
"Definition:Vector Subspace",
"Set of Linear Subspaces is Closed under Intersection",
"Definition:Vector Subspace",
"Codimension of Whole Vector Space",
"Kernel of Non-Zero Linear Functional has Codimensio... |
proofwiki-23032 | Kottman's Improvement of Riesz' Lemma | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be an infinite-dimensional normed vector space over $\GF$.
Then there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ such that $\norm {x_n}_X = 1$ for each $n \in \N$ and:
:$\norm {x_n - x_m}_X > 1$ for each $n \ne m$.
That is, the Kottman c... | We firstly take $\GF = \R$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $x_1 \in X$ have $\norm {x_n}_X = 1$.
From Existence of Support Functional, there exists $f_1 \in X^\ast$ such that $\norm {f_1}_{X^\ast} = 1$ and:
:$\map {f_1} {x_1} ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Then there exists a [[Definition:Sequence|sequence]] $\sequence {x_n}_{n \mathop \in \N}$ such that $\no... | We firstly take $\GF = \R$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $x_1 \in X$ have $\norm {x_n}_X = 1$.
From [[Existence of Support Functional]], there exists $f_1 \in X^\ast$ such that $\norm {f_1... | Kottman's Improvement of Riesz' Lemma | https://proofwiki.org/wiki/Kottman's_Improvement_of_Riesz'_Lemma | https://proofwiki.org/wiki/Kottman's_Improvement_of_Riesz'_Lemma | [
"Normed Vector Spaces"
] | [
"Definition:Infinite-Dimensional Vector Space",
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Kottman Constant of Banach Space"
] | [
"Definition:Normed Dual Space",
"Existence of Support Functional",
"Definition:Sequence",
"Definition:Linearly Independent",
"Definition:Linearly Independent",
"Condition for Linear Dependence of Linear Functionals in terms of Kernel",
"Intersection of Finitely Many Kernels of Linear Functionals in Infi... |
proofwiki-23033 | Duality Mapping of Dual System of Normed Vector Space with Normed Dual is Jointly Continuous | 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 $\struct {X \times X^\ast, \norm {\, \cdot \,}_{X \times X^\ast} }$ be the direct product... | Let $\sequence {\tuple {x_n, f_n} }_{n \mathop \in \N}$ be a sequence in $X \times X^\ast$ converging to $\tuple {x, f}$.
We then have:
{{begin-eqn}}
{{eqn | l = \cmod {\innerprod {x_n} {f_n} - \innerprod x f}
| r = \cmod {\innerprod {x_n} {f_n} - \innerprod {x_n} f + \innerprod {x_n} f - \innerprod x f}
}}
{{eqn | o... | 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 $\struct {X \times... | Let $\sequence {\tuple {x_n, f_n} }_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X \times X^\ast$ [[Definition:Convergent Sequence|converging]] to $\tuple {x, f}$.
We then have:
{{begin-eqn}}
{{eqn | l = \cmod {\innerprod {x_n} {f_n} - \innerprod x f}
| r = \cmod {\innerprod {x_n} {f_n} - \innerprod... | Duality Mapping of Dual System of Normed Vector Space with Normed Dual is Jointly Continuous | https://proofwiki.org/wiki/Duality_Mapping_of_Dual_System_of_Normed_Vector_Space_with_Normed_Dual_is_Jointly_Continuous | https://proofwiki.org/wiki/Duality_Mapping_of_Dual_System_of_Normed_Vector_Space_with_Normed_Dual_is_Jointly_Continuous | [
"Dual Systems",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Direct Product Norm",
"Definition:Continuous Mapping"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence",
"Triangle Inequality",
"Fundamental Property of Norm on Bounded Linear Functional",
"Convergent Sequence in Normed Vector Space is Bounded",
"Convergence in Direct Product Norm",
"Definition:Sequential Continuity",
"Sequential Continuity is Equi... |
proofwiki-23034 | Pointwise Supremum of Family of Lower Semicontinuous Functions is Lower Semicontinuous | Let $\struct {X, \tau}$ be a topological space.
Let $\family {f_\alpha}_{\alpha \mathop \in I}$ be a set of lower semicontinuous functions $f_\alpha : X \to \overline \R$.
Define $f : X \to \overline \R$ by:
:$\ds \map f x = \limsup_{\alpha \mathop \in I} \map {f_\alpha} x$ for each $x \in X$.
Then $f$ is lower semicon... | For $t \in \R$, we have:
:$\ds \limsup_{\alpha \mathop \in I} \map {f_\alpha} x \le t$
{{iff}}:
:$\map {f_\alpha} x \le t$ for each $\alpha \in I$.
Hence:
:$\ds \set {x \in X : \map f x \le t} = \bigcap_{\alpha \mathop \in I} \set {x \in X : \map {f_\alpha} x \le t}$
Since $f_\alpha$ is lower semicontinuous, we have:
:... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\family {f_\alpha}_{\alpha \mathop \in I}$ be a [[Definition:Set|set]] of [[Definition:Lower Semicontinuous|lower semicontinuous functions]] $f_\alpha : X \to \overline \R$.
Define $f : X \to \overline \R$ by:
:$\ds \map f x = \lims... | For $t \in \R$, we have:
:$\ds \limsup_{\alpha \mathop \in I} \map {f_\alpha} x \le t$
{{iff}}:
:$\map {f_\alpha} x \le t$ for each $\alpha \in I$.
Hence:
:$\ds \set {x \in X : \map f x \le t} = \bigcap_{\alpha \mathop \in I} \set {x \in X : \map {f_\alpha} x \le t}$
Since $f_\alpha$ is [[Definition:Lower Semicontinu... | Pointwise Supremum of Family of Lower Semicontinuous Functions is Lower Semicontinuous | https://proofwiki.org/wiki/Pointwise_Supremum_of_Family_of_Lower_Semicontinuous_Functions_is_Lower_Semicontinuous | https://proofwiki.org/wiki/Pointwise_Supremum_of_Family_of_Lower_Semicontinuous_Functions_is_Lower_Semicontinuous | [
"Lower Semicontinuous",
"Lower Semicontinuity",
"Lower Semicontinuity"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Lower Semicontinuous",
"Definition:Lower Semicontinuous"
] | [
"Definition:Lower Semicontinuous",
"Definition:Closed Set",
"Characterization of Lower Semicontinuity",
"Definition:Intersection",
"Definition:Closed Set",
"Characterization of Lower Semicontinuity",
"Definition:Lower Semicontinuous"
] |
proofwiki-23035 | Negative of Limit Superior of Real Net is Limit Inferior | Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a net valued in $\R$.
Then:
:$\ds -\limsup_{\lambda \mathop \in \Lambda} x_\lambda = \liminf_{\lambda \mathop \in \Lambda} \paren {-x_\lambda}$ | We have:
{{begin-eqn}}
{{eqn | l = -\limsup_{\lambda \mathop \in \Lambda} x_\lambda
| r = -\inf_{\lambda_0 \mathop \in \Lambda} \sup_{\lambda \succeq \lambda_0} x_\lambda
| c = {{Defof|Limit Superior of Net}}
}}
{{eqn | r = \sup_{\lambda \mathop \in \Lambda} \paren {-\sup_{\lambda \succeq \lambda_0} x_\lambda}
| ... | Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a [[Definition:Net (Set Theory)|net]] valued in $\R$.
Then:
:$\ds -\limsup_{\lambda \mathop \in \Lambda} x_\lambda = \liminf_{\lambda \mathop \in \Lambda} \paren {-x_\lambda}$ | We have:
{{begin-eqn}}
{{eqn | l = -\limsup_{\lambda \mathop \in \Lambda} x_\lambda
| r = -\inf_{\lambda_0 \mathop \in \Lambda} \sup_{\lambda \succeq \lambda_0} x_\lambda
| c = {{Defof|Limit Superior of Net}}
}}
{{eqn | r = \sup_{\lambda \mathop \in \Lambda} \paren {-\sup_{\lambda \succeq \lambda_0} x_\lambda}
| ... | Negative of Limit Superior of Real Net is Limit Inferior | https://proofwiki.org/wiki/Negative_of_Limit_Superior_of_Real_Net_is_Limit_Inferior | https://proofwiki.org/wiki/Negative_of_Limit_Superior_of_Real_Net_is_Limit_Inferior | [
"Limits Superior",
"Limits Inferior"
] | [
"Definition:Directed Preordering",
"Definition:Net (Set Theory)"
] | [
"Negative of Infimum is Supremum of Negatives",
"Negative of Supremum is Infimum of Negatives",
"Category:Limits Superior",
"Category:Limits Inferior"
] |
proofwiki-23036 | Real-Valued Lower Semicontinuous Function on Compact Space attains Minimum with Compact Set of Minimizers | Let $\struct {X, \tau}$ be a compact topological space.
Let $f : X \to \R$ be a lower semicontinuous function.
Then there exists $x_0 \in X$ such that:
:$\ds \map f {x_0} = \inf_{x \mathop \in X} \map f x$
and $\ds f^{-1} \sqbrk {\inf_{x \mathop \in X} \map f x}$ is compact. | Let $A = f \sqbrk X$.
For each $c \in A$, define:
:$F_c = \set {x \in X : \map f x \le c}$
Since $f$ is lower semicontinuous, $F_c$ is closed for each $c \in A$.
Further, for $c_1, \ldots, c_n \in A$, we have:
{{begin-eqn}}
{{eqn | l = \bigcap_{j \mathop = 1}^n F_{c_j}
| r = \bigcap_{j \mathop = 1}^n \set {x \in X :... | Let $\struct {X, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $f : X \to \R$ be a [[Definition:Lower Semicontinuous|lower semicontinuous function]].
Then there exists $x_0 \in X$ such that:
:$\ds \map f {x_0} = \inf_{x \mathop \in X} \map f x$
and $\ds f^{-1} \sqbrk {\inf_{x \... | Let $A = f \sqbrk X$.
For each $c \in A$, define:
:$F_c = \set {x \in X : \map f x \le c}$
Since $f$ is [[Definition:Lower Semicontinuous|lower semicontinuous]], $F_c$ is [[Definition:Closed Set|closed]] for each $c \in A$.
Further, for $c_1, \ldots, c_n \in A$, we have:
{{begin-eqn}}
{{eqn | l = \bigcap_{j \mathop... | Real-Valued Lower Semicontinuous Function on Compact Space attains Minimum with Compact Set of Minimizers | https://proofwiki.org/wiki/Real-Valued_Lower_Semicontinuous_Function_on_Compact_Space_attains_Minimum_with_Compact_Set_of_Minimizers | https://proofwiki.org/wiki/Real-Valued_Lower_Semicontinuous_Function_on_Compact_Space_attains_Minimum_with_Compact_Set_of_Minimizers | [
"Lower Semicontinuity",
"Compact Topological Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:Lower Semicontinuous",
"Definition:Compact Topological Space"
] | [
"Definition:Lower Semicontinuous",
"Definition:Closed Set",
"Definition:Finite Intersection Property",
"Definition:Finite Intersection Property",
"Intersection of Closed Subsets satisfying Finite Intersection Property in Compact Space",
"Definition:Compact Topological Space",
"Definition:Closed Set",
... |
proofwiki-23037 | Real-Valued Upper Semicontinuous Function on Compact Space attains Maximum with Compact Set of Maximizers | Let $\struct {X, \tau}$ be a compact topological space.
Let $f : X \to \R$ be a upper semicontinuous function.
Then there exists $x_0 \in X$ such that:
:$\ds \map f {x_0} = \sup_{x \mathop \in X} \map f x$
and $\ds f^{-1} \sqbrk {\sup_{x \mathop \in X} \map f x}$ is compact. | From Function is Upper Semicontinuous iff Negative is Lower Semicontinuous, $-f$ is lower semicontinuous.
From Real-Valued Lower Semicontinuous Function on Compact Space attains Minimum with Compact Set of Minimizers, there exists $x_0 \in X$ such that:
:$\ds -\map f {x_0} = \inf_{x \mathop \in X} \paren {-\map f x}$
F... | Let $\struct {X, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $f : X \to \R$ be a [[Definition:Upper Semicontinuous|upper semicontinuous function]].
Then there exists $x_0 \in X$ such that:
:$\ds \map f {x_0} = \sup_{x \mathop \in X} \map f x$
and $\ds f^{-1} \sqbrk {\sup_{x \... | From [[Function is Upper Semicontinuous iff Negative is Lower Semicontinuous]], $-f$ is [[Definition:Lower Semicontinuous|lower semicontinuous]].
From [[Real-Valued Lower Semicontinuous Function on Compact Space attains Minimum with Compact Set of Minimizers]], there exists $x_0 \in X$ such that:
:$\ds -\map f {x_0} =... | Real-Valued Upper Semicontinuous Function on Compact Space attains Maximum with Compact Set of Maximizers | https://proofwiki.org/wiki/Real-Valued_Upper_Semicontinuous_Function_on_Compact_Space_attains_Maximum_with_Compact_Set_of_Maximizers | https://proofwiki.org/wiki/Real-Valued_Upper_Semicontinuous_Function_on_Compact_Space_attains_Maximum_with_Compact_Set_of_Maximizers | [
"Upper Semicontinuity"
] | [
"Definition:Compact Topological Space",
"Definition:Upper Semicontinuous",
"Definition:Compact Topological Space"
] | [
"Function is Upper Semicontinuous iff Negative is Lower Semicontinuous",
"Definition:Lower Semicontinuous",
"Real-Valued Lower Semicontinuous Function on Compact Space attains Minimum with Compact Set of Minimizers",
"Negative of Supremum is Infimum of Negatives",
"Real-Valued Lower Semicontinuous Function ... |
proofwiki-23038 | Evaluation Mapping of Infinite-Dimensional Normed Vector Space is not Jointly Continuous with Normed Vector Space given Weak Topology | 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 of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w$ be the weak topology of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $\struct {X, w} \times \struct {X^\ast, \norm {\, \cdot... | Let $\struct {\FF, \subseteq}$ be the set of non-empty finite subsets of $X^\ast$ equipped with set inclusion.
From Set of Finite Subsets is Directed Set with Set Inclusion, $\struct {\FF, \subseteq}$ is directed.
From Intersection of Finitely Many Kernels of Linear Functionals in Infinite-Dimensional Vector Space is I... | 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]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w$ be the [[Definition:Weak Topology|weak topology]] o... | Let $\struct {\FF, \subseteq}$ be the [[Definition:Set|set]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Subset|finite subsets]] of $X^\ast$ equipped with [[Definition:Set Inclusion|set inclusion]].
From [[Set of Finite Subsets is Directed Set with Set Inclusion]], $\struct {\FF, \subseteq}$ is [[Def... | Evaluation Mapping of Infinite-Dimensional Normed Vector Space is not Jointly Continuous with Normed Vector Space given Weak Topology | https://proofwiki.org/wiki/Evaluation_Mapping_of_Infinite-Dimensional_Normed_Vector_Space_is_not_Jointly_Continuous_with_Normed_Vector_Space_given_Weak_Topology | https://proofwiki.org/wiki/Evaluation_Mapping_of_Infinite-Dimensional_Normed_Vector_Space_is_not_Jointly_Continuous_with_Normed_Vector_Space_given_Weak_Topology | [
"Normed Vector Spaces",
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Initial Topology",
"Definition:Direct Product of Vector Spaces",
"Definition:Product Topology",
"Definition:Continuous Mapping (Topology)/Point"
] | [
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Finite Subset",
"Definition:Subset",
"Set of Finite Subsets is Directed Set with Set Inclusion",
"Definition:Directed Preordering",
"Intersection of Finitely Many Kernels of Linear Functionals in Infinite-Dimensional Vector Space is Infinite-Dime... |
proofwiki-23039 | Evaluation Mapping of Infinite-Dimensional Normed Vector Space is not Jointly Continuous with Normed Dual Space given Weak-* Topology | 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 of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w^\ast$ be the weak-$\ast$ topology on $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$.
Let $\struct {X, \norm {\, \cdot \... | Let $\struct {\FF, \subseteq}$ be the set of non-empty finite subsets of $X$ equipped with set inclusion.
From Set of Finite Subsets is Directed Set with Set Inclusion, $\struct {\FF, \subseteq}$ is directed.
For each $x \in X$, define $\hat x : X^\ast \to \GF$ by:
:$\map {\hat x} f = \map f x$ for each $f \in X^\ast$.... | 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]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Let $w^\ast$ be the [[Definition:Weak-* Topology|weak-$\ast... | Let $\struct {\FF, \subseteq}$ be the [[Definition:Set|set]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Subset|finite subsets]] of $X$ equipped with [[Definition:Set Inclusion|set inclusion]].
From [[Set of Finite Subsets is Directed Set with Set Inclusion]], $\struct {\FF, \subseteq}$ is [[Definiti... | Evaluation Mapping of Infinite-Dimensional Normed Vector Space is not Jointly Continuous with Normed Dual Space given Weak-* Topology | https://proofwiki.org/wiki/Evaluation_Mapping_of_Infinite-Dimensional_Normed_Vector_Space_is_not_Jointly_Continuous_with_Normed_Dual_Space_given_Weak-*_Topology | https://proofwiki.org/wiki/Evaluation_Mapping_of_Infinite-Dimensional_Normed_Vector_Space_is_not_Jointly_Continuous_with_Normed_Dual_Space_given_Weak-*_Topology | [
"Normed Vector Spaces",
"Weak-* Topologies"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Weak-* Topology",
"Definition:Direct Product of Vector Spaces",
"Definition:Product Topology",
"Definition:Continuous Mapping (Topology)/Point"
] | [
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Finite Subset",
"Definition:Subset",
"Set of Finite Subsets is Directed Set with Set Inclusion",
"Definition:Directed Preordering",
"Intersection of Finitely Many Kernels of Linear Functionals in Infinite-Dimensional Vector Space is Infinite-Dime... |
proofwiki-23040 | Existence of Locally Convex Topology Consistent with Dual System | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Then there exists a locally convex topology on $E$ consistent with $\innerprod E F$. | For each $f \in F$, define $\pi_f : E \to \GF$ by:
:$\map {\pi_f} x = \innerprod x f$ for each $x \in E$.
Let $\tau$ be the initial topology on $E$ induced by $\pi_f$.
This is precisely the weak topology $\map \sigma {E, F}$.
Let $\struct {E, \tau}^\ast$ be the topological dual space of $\struct {E, \tau}$.
We aim to ... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Then there exists a [[Definition:Locally Convex Topology Consistent with Dual System|locally convex topology on $E$ consistent with $\innerprod E F$]]. | For each $f \in F$, define $\pi_f : E \to \GF$ by:
:$\map {\pi_f} x = \innerprod x f$ for each $x \in E$.
Let $\tau$ be the [[Definition:Initial Topology|initial topology]] on $E$ induced by $\pi_f$.
This is precisely the [[Definition:Weak Topology Induced by Dual System|weak topology]] $\map \sigma {E, F}$.
Let $\... | Existence of Locally Convex Topology Consistent with Dual System | https://proofwiki.org/wiki/Existence_of_Locally_Convex_Topology_Consistent_with_Dual_System | https://proofwiki.org/wiki/Existence_of_Locally_Convex_Topology_Consistent_with_Dual_System | [
"Dual Systems"
] | [
"Definition:Dual System",
"Definition:Locally Convex Topology Consistent with Dual System"
] | [
"Definition:Initial Topology",
"Definition:Weak Topology Induced by Dual System",
"Definition:Topological Dual Space",
"Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex",
"Definition:Locally Convex Space",
"Definition:Locally Convex Space/Standard Topology",
"Continuity... |
proofwiki-23041 | Duality Mapping of Dual System is Jointly Continuous when Dual Restricted to Absolute Polar of Open Neighborhood of Origin | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a dual system over $\GF$.
Let $\tau$ be a locally convex topology on $E$ consistent with $\innerprod E F$.
Let $\map \sigma {F, E}$ be the weak-$\ast$ topology induced by $\innerprod E F$.
Let $V$ be an open neighborhood of ${\mathbf 0}_E$ in $\struct {E, \tau}$.
Le... | We use Characterization of Continuity in terms of Nets.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $\family {\tuple {x_\lambda, f_\lambda} }_{\lambda \mathop \in \Lambda}$ be a net in $E \times V^\circ$ converging to $\struct {x, f}$.
We aim to show that $\innerprod {x_\lambda} {f_\lambda} \to \innerprod x... | Let $\GF \in \set {\R, \C}$.
Let $\innerprod E F$ be a [[Definition:Dual System|dual system]] over $\GF$.
Let $\tau$ be a [[Definition:Locally Convex Topology Consistent with Dual System|locally convex topology on $E$ consistent with $\innerprod E F$]].
Let $\map \sigma {F, E}$ be the [[Definition:Weak-* Topology In... | We use [[Characterization of Continuity in terms of Nets]].
Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $\family {\tuple {x_\lambda, f_\lambda} }_{\lambda \mathop \in \Lambda}$ be a [[Definition:Net (Set Theory)|net]] in $E \times V^\circ$ [[Definition:Convergent Net|converging... | Duality Mapping of Dual System is Jointly Continuous when Dual Restricted to Absolute Polar of Open Neighborhood of Origin | https://proofwiki.org/wiki/Duality_Mapping_of_Dual_System_is_Jointly_Continuous_when_Dual_Restricted_to_Absolute_Polar_of_Open_Neighborhood_of_Origin | https://proofwiki.org/wiki/Duality_Mapping_of_Dual_System_is_Jointly_Continuous_when_Dual_Restricted_to_Absolute_Polar_of_Open_Neighborhood_of_Origin | [
"Dual Systems",
"Absolute Polars"
] | [
"Definition:Dual System",
"Definition:Locally Convex Topology Consistent with Dual System",
"Definition:Weak-* Topology Induced by Dual System",
"Definition:Open Neighborhood",
"Definition:Absolute Polar",
"Definition:Direct Product of Vector Spaces",
"Definition:Product Topology",
"Definition:Continu... | [
"Characterization of Continuity in terms of Nets",
"Definition:Directed Preordering",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Characterization of Convergence of Net in Product Topology",
"Triangle Inequality",
"Dilation of Open Set in Topological Vector Space is Open",
"Definition... |
proofwiki-23042 | Metric Projection Fixes Points of Convex Set | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\GF$.
Let $\norm {\, \cdot \,}_\HH$ be the inner product norm.
Let $\pi_C : \HH \to C$ be the metric projection of $\HH$ onto $C$.
Then:
:$\map {\pi_C} x = x$ for each $x \in C$. | Let $x \in C$.
We have:
:$\ds \inf_{z \mathop \in C} \norm {x - z}_\HH \le \norm {x - x}_\HH = 0$
Hence:
:$\ds \inf_{z \mathop \in C} \norm {x - z}_\HH = 0$
Hence from Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space and the definition of the metric projection we have $\map {\pi_C} x = x$.
{{qe... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\GF$.
Let $\norm {\, \cdot \,}_\HH$ be the [[Definition:Inner Product Norm|inner product norm]].
Let $\pi_C : \HH \to C$ be the [[Definition:Metric Projection|metric projection]] of $\HH$ on... | Let $x \in C$.
We have:
:$\ds \inf_{z \mathop \in C} \norm {x - z}_\HH \le \norm {x - x}_\HH = 0$
Hence:
:$\ds \inf_{z \mathop \in C} \norm {x - z}_\HH = 0$
Hence from [[Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space]] and the definition of the [[Definition:Metric Projection|metric project... | Metric Projection Fixes Points of Convex Set | https://proofwiki.org/wiki/Metric_Projection_Fixes_Points_of_Convex_Set | https://proofwiki.org/wiki/Metric_Projection_Fixes_Points_of_Convex_Set | [
"Metric Projections"
] | [
"Definition:Hilbert Space",
"Definition:Inner Product Norm",
"Definition:Metric Projection"
] | [
"Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space",
"Definition:Metric Projection"
] |
proofwiki-23043 | Metric Projection of Point Lies on Boundary of Convex Set | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\GF$.
Let $\norm {\, \cdot \,}_\HH$ be the inner product norm.
Let $\pi_C : \HH \to C$ be the metric projection of $\HH$ onto $C$.
Let $\partial C$ be the topological boundary of $C$.
Let $x \in \HH \setminus C$.
Then $\... | Since $x \in \HH \setminus C$, we have $x \ne \map {\pi_C} x$.
Hence $\norm {x - \map {\pi_C} x}_\HH \ne 0$ from {{NormAxiomVector|1}}.
Suppose that $\map {\pi_C} x \in \Int C$, where $\Int C$ is the interior of $C$.
For $\lambda \in \closedint 0 1$, define:
:$x_\lambda = \lambda x + \paren {1 - \lambda} \map {\pi_C} x... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\GF$.
Let $\norm {\, \cdot \,}_\HH$ be the [[Definition:Inner Product Norm|inner product norm]].
Let $\pi_C : \HH \to C$ be the [[Definition:Metric Projection|metric projection]] of $\HH$ on... | Since $x \in \HH \setminus C$, we have $x \ne \map {\pi_C} x$.
Hence $\norm {x - \map {\pi_C} x}_\HH \ne 0$ from {{NormAxiomVector|1}}.
Suppose that $\map {\pi_C} x \in \Int C$, where $\Int C$ is the [[Definition:Interior (Topology)|interior]] of $C$.
For $\lambda \in \closedint 0 1$, define:
:$x_\lambda = \lambda x... | Metric Projection of Point Lies on Boundary of Convex Set | https://proofwiki.org/wiki/Metric_Projection_of_Point_Lies_on_Boundary_of_Convex_Set | https://proofwiki.org/wiki/Metric_Projection_of_Point_Lies_on_Boundary_of_Convex_Set | [
"Metric Projections"
] | [
"Definition:Hilbert Space",
"Definition:Inner Product Norm",
"Definition:Metric Projection",
"Definition:Boundary (Topology)"
] | [
"Definition:Interior (Topology)"
] |
proofwiki-23044 | Bound on Inner Product of Differences with Metric Projection | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\R$.
Let $\norm {\, \cdot \,}_\HH$ be the inner product norm.
Let $C \subseteq \HH$ be a non-empty closed convex set.
Let $\pi_C : \HH \to C$ be the metric projection of $\HH$ onto $C$.
Let $x \in \HH$ and $y \in C$.
Then we have:
:$\innerprod {x - \... | Let $\alpha \in \openint 0 1$.
We have:
:$\alpha y + \paren {1 - \alpha} \map {\pi_C} x \in C$
since $C$ is convex and $y, \map {\pi_C} x \in C$.
We then have:
{{begin-eqn}}
{{eqn | l = \norm {x - \map {\pi_C} x}_\HH^2
| r = \inf_{z \mathop \in C} \norm {x - z}_\HH^2
| c = {{Defof|Metric Projection}}
}}
{{eqn | o ... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\R$.
Let $\norm {\, \cdot \,}_\HH$ be the [[Definition:Inner Product Norm|inner product norm]].
Let $C \subseteq \HH$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Closed Set|closed]] [[Definition:Convex Set (... | Let $\alpha \in \openint 0 1$.
We have:
:$\alpha y + \paren {1 - \alpha} \map {\pi_C} x \in C$
since $C$ is [[Definition:Convex Set (Vector Space)|convex]] and $y, \map {\pi_C} x \in C$.
We then have:
{{begin-eqn}}
{{eqn | l = \norm {x - \map {\pi_C} x}_\HH^2
| r = \inf_{z \mathop \in C} \norm {x - z}_\HH^2
| c ... | Bound on Inner Product of Differences with Metric Projection | https://proofwiki.org/wiki/Bound_on_Inner_Product_of_Differences_with_Metric_Projection | https://proofwiki.org/wiki/Bound_on_Inner_Product_of_Differences_with_Metric_Projection | [
"Metric Projections",
"Bound on Inner Product of Differences with Metric Projection"
] | [
"Definition:Hilbert Space",
"Definition:Inner Product Norm",
"Definition:Non-Empty Set",
"Definition:Closed Set",
"Definition:Convex Set (Vector Space)",
"Definition:Metric Projection"
] | [
"Definition:Convex Set (Vector Space)"
] |
proofwiki-23045 | Metric Projection onto Closed Convex Set is 1-Lipschitz | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\R$.
Let $\norm {\, \cdot \,}_\HH$ be the inner product norm.
Let $C \subseteq \HH$ be a non-empty closed convex set.
Let $\pi_C : \HH \to C$ be the metric projection of $\HH$ onto $C$.
Then $\pi_C$ is $1$-Lipschitz. | Let $x, y \in \HH$.
From Bound on Inner Product of Differences with Metric Projection, we have:
:$\innerprod {x - \map {\pi_C} x} {\map {\pi_C} y - \map {\pi_C} x} \le 0$
since $\map {\pi_C} y \in C$.
Similarly:
:$\innerprod {y - \map {\pi_C} y} {\map {\pi_C} x - \map {\pi_C} y} \le 0$
since $\map {\pi_C} x \in C$.
Tha... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\R$.
Let $\norm {\, \cdot \,}_\HH$ be the [[Definition:Inner Product Norm|inner product norm]].
Let $C \subseteq \HH$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Closed Set|closed]] [[Definition:Convex Set (... | Let $x, y \in \HH$.
From [[Bound on Inner Product of Differences with Metric Projection]], we have:
:$\innerprod {x - \map {\pi_C} x} {\map {\pi_C} y - \map {\pi_C} x} \le 0$
since $\map {\pi_C} y \in C$.
Similarly:
:$\innerprod {y - \map {\pi_C} y} {\map {\pi_C} x - \map {\pi_C} y} \le 0$
since $\map {\pi_C} x \in C... | Metric Projection onto Closed Convex Set is 1-Lipschitz | https://proofwiki.org/wiki/Metric_Projection_onto_Closed_Convex_Set_is_1-Lipschitz | https://proofwiki.org/wiki/Metric_Projection_onto_Closed_Convex_Set_is_1-Lipschitz | [
"Metric Projections",
"Lipschitz Continuity"
] | [
"Definition:Hilbert Space",
"Definition:Inner Product Norm",
"Definition:Non-Empty Set",
"Definition:Closed Set",
"Definition:Convex Set (Vector Space)",
"Definition:Metric Projection",
"Definition:Lipschitz Continuity"
] | [
"Bound on Inner Product of Differences with Metric Projection",
"Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces"
] |
proofwiki-23046 | Topological Dual Space of Product of Topological Vector Spaces | Let $\struct {K, \tau_K}$ be a topological field.
Let $\struct {X, \tau_X}$ and $\struct {Y, \tau_Y}$ be topological vector spaces over $K$.
Let $\struct {X \times Y, \tau}$ be the direct product of $X$ and $Y$ equipped with the product topology.
Let $\paren {X \times Y}^\ast$, $X^\ast$ and $Y^\ast$ denote the topologi... | === Necessary Condition ===
Let $\phi \in \paren {X \times Y}^\ast$.
For each $\tuple {x, y} \in X \times Y$, we have:
:$\map \phi {x, y} = \map \phi {x, {\mathbf 0}_Y} + \map \phi { {\mathbf 0}_X, y}$
since $\phi$ is linear.
Define $\phi_X : X \to K$ by:
:$\map {\phi_X} x = \map \phi {x, {\mathbf 0}_Y}$ for each $x \i... | Let $\struct {K, \tau_K}$ be a [[Definition:Topological Field|topological field]].
Let $\struct {X, \tau_X}$ and $\struct {Y, \tau_Y}$ be [[Definition:Topological Vector Space|topological vector spaces]] over $K$.
Let $\struct {X \times Y, \tau}$ be the [[Definition:Direct Product of Vector Spaces|direct product]] of... | === Necessary Condition ===
Let $\phi \in \paren {X \times Y}^\ast$.
For each $\tuple {x, y} \in X \times Y$, we have:
:$\map \phi {x, y} = \map \phi {x, {\mathbf 0}_Y} + \map \phi { {\mathbf 0}_X, y}$
since $\phi$ is [[Definition:Linear Functional|linear]].
Define $\phi_X : X \to K$ by:
:$\map {\phi_X} x = \map \ph... | Topological Dual Space of Product of Topological Vector Spaces | https://proofwiki.org/wiki/Topological_Dual_Space_of_Product_of_Topological_Vector_Spaces | https://proofwiki.org/wiki/Topological_Dual_Space_of_Product_of_Topological_Vector_Spaces | [
"Topological Vector Spaces",
"Direct Product of Vector Spaces",
"Topological Dual Space of Product of Topological Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Product Topology",
"Definition:Topological Dual Space",
"Definition:Isomorphism",
"Definition:Direct Product of Vector Spaces"
] | [
"Definition:Linear Functional",
"Horizontal Section of Linear Transformation is Linear Transformation",
"Definition:Linear Functional",
"Vertical Section of Linear Transformation is Linear Transformation",
"Definition:Linear Functional",
"Horizontal Section of Continuous Function is Continuous",
"Defini... |
proofwiki-23047 | Intersection of Epigraphs is Epigraph of Supremum | Let $X$ be a set.
Let $\family {f_\alpha}_{\alpha \mathop \in I}$ be an $I$-indexed family of functions $f_\alpha : X \to \overline \R$.
Let $\map {\operatorname {epi} } {f_\alpha}$ be the epigraph of $f_\alpha$ for each $\alpha \in I$.
Let $\ds \map {\operatorname {epi} } {\sup_{\alpha \mathop \in I} f_\alpha}$ be the... | We have:
{{begin-eqn}}
{{eqn | l = \bigcap_{\alpha \mathop \in I} \map {\operatorname {epi} } {f_\alpha}
| r = \bigcap_{\alpha \mathop \in I} \set {\tuple {x, c} \in X \times \R : \map {f_\alpha} x \le c}
| c = {{Defof|Epigraph}}
}}
{{eqn | r = \set {\tuple {x, c} \in X \times \R : \map {f_\alpha} x \le c \text { f... | Let $X$ be a [[Definition:Set|set]].
Let $\family {f_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family of Sets|$I$-indexed family]] of [[Definition:Function|functions]] $f_\alpha : X \to \overline \R$.
Let $\map {\operatorname {epi} } {f_\alpha}$ be the [[Definition:Epigraph|epigraph]] of $f_\alpha$ f... | We have:
{{begin-eqn}}
{{eqn | l = \bigcap_{\alpha \mathop \in I} \map {\operatorname {epi} } {f_\alpha}
| r = \bigcap_{\alpha \mathop \in I} \set {\tuple {x, c} \in X \times \R : \map {f_\alpha} x \le c}
| c = {{Defof|Epigraph}}
}}
{{eqn | r = \set {\tuple {x, c} \in X \times \R : \map {f_\alpha} x \le c \text { f... | Intersection of Epigraphs is Epigraph of Supremum | https://proofwiki.org/wiki/Intersection_of_Epigraphs_is_Epigraph_of_Supremum | https://proofwiki.org/wiki/Intersection_of_Epigraphs_is_Epigraph_of_Supremum | [
"Epigraphs"
] | [
"Definition:Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Function",
"Definition:Epigraph",
"Definition:Epigraph",
"Definition:Pointwise Supremum of Extended Real-Valued Functions"
] | [
"Category:Epigraphs"
] |
proofwiki-23048 | Pointwise Infimum of Family of Concave Real Functions is Concave | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $C \subseteq X$ be a non-empty convex set.
Let $\family {f_\alpha}_{\alpha \mathop \in I}$ be an $I$-indexed family of concave functions $f_\alpha : C \to \overline \R$.
Let:
:$\ds f = \inf_{\alpha \mathop \in I} f_\alpha$
be the pointwise infimum o... | Let $\operatorname {hypo}$ denote hypograph.
From Intersection of Hypographs is Hypograph of Infimum, we have:
:$\ds \map {\operatorname {hypo} } f = \bigcap_{\alpha \mathop \in I} \map {\operatorname {hypo} } {f_\alpha}$
From Function is Concave iff Hypograph is Convex, we have that:
:$\map {\operatorname {hypo} } {f... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $C \subseteq X$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Convex Set (Vector Space)|convex set]].
Let $\family {f_\alpha}_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family of Sets|$I$-indexed fam... | Let $\operatorname {hypo}$ denote [[Definition:Hypograph|hypograph]].
From [[Intersection of Hypographs is Hypograph of Infimum]], we have:
:$\ds \map {\operatorname {hypo} } f = \bigcap_{\alpha \mathop \in I} \map {\operatorname {hypo} } {f_\alpha}$
From [[Function is Concave iff Hypograph is Convex]], we have that... | Pointwise Infimum of Family of Concave Real Functions is Concave | https://proofwiki.org/wiki/Pointwise_Infimum_of_Family_of_Concave_Real_Functions_is_Concave | https://proofwiki.org/wiki/Pointwise_Infimum_of_Family_of_Concave_Real_Functions_is_Concave | [
"Epigraphs",
"Hypographs",
"Convex Real Functions",
"Concave Real Functions",
"Hypographs",
"Concave Real Functions"
] | [
"Definition:Vector Space",
"Definition:Non-Empty Set",
"Definition:Convex Set (Vector Space)",
"Definition:Indexing Set/Family of Sets",
"Definition:Concave Real Function/Vector Space",
"Definition:Pointwise Infimum of Extended Real-Valued Functions",
"Definition:Concave Real Function/Vector Space"
] | [
"Definition:Hypograph",
"Intersection of Hypographs is Hypograph of Infimum",
"Function is Concave iff Hypograph is Convex",
"Definition:Convex Set (Vector Space)",
"Intersection of Convex Sets is Convex Set (Vector Spaces)",
"Definition:Convex Set (Vector Space)",
"Function is Concave iff Hypograph is ... |
proofwiki-23049 | Affine Functional is Convex and Concave | Let $X$ be a vector space over $\R$.
Let $f : X \to \R$ be an affine functional on $X$.
Then $f$ is convex and concave. | Let $g : X \to \R$ be a linear functional be such that:
:$\map f x = \map g x + c$ for all $x \in X$
for some $c \in \R$.
Let $x, y \in X$ and $t \in \closedint 0 1$.
Then we have:
{{begin-eqn}}
{{eqn | l = t \map f x + \paren {1 - t} \map f y
| r = t \paren {\map g x + c} + \paren {1 - t} \paren {\map g y + c}
}}
{... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\R$.
Let $f : X \to \R$ be an [[Definition:Affine Functional|affine functional]] on $X$.
Then $f$ is [[Definition:Convex Real Function/Vector Space|convex]] and [[Definition:Concave Real Function/Vector Space|concave]]. | Let $g : X \to \R$ be a [[Definition:Linear Functional|linear functional]] be such that:
:$\map f x = \map g x + c$ for all $x \in X$
for some $c \in \R$.
Let $x, y \in X$ and $t \in \closedint 0 1$.
Then we have:
{{begin-eqn}}
{{eqn | l = t \map f x + \paren {1 - t} \map f y
| r = t \paren {\map g x + c} + \paren... | Affine Functional is Convex and Concave | https://proofwiki.org/wiki/Affine_Functional_is_Convex_and_Concave | https://proofwiki.org/wiki/Affine_Functional_is_Convex_and_Concave | [
"Affine Functionals",
"Convex Real Functions",
"Concave Real Functions"
] | [
"Definition:Vector Space",
"Definition:Affine Functional",
"Definition:Convex Real Function/Vector Space",
"Definition:Concave Real Function/Vector Space"
] | [
"Definition:Linear Functional",
"Definition:Convex Real Function/Vector Space",
"Definition:Concave Real Function/Vector Space",
"Category:Affine Functionals",
"Category:Convex Real Functions",
"Category:Concave Real Functions"
] |
proofwiki-23050 | Concave Envelope is Concave Majorant of Function | Let $\struct {X, \tau}$ be a topological vector space over $\R$.
Let $C \subseteq X$ be a non-empty closed convex set.
Let $f : C \to \R$ be a function.
Let $\hat f : C \to \overline \R$ be the concave envelope of $f$.
Then $\hat f$ is concave and $f \le \hat f$. | We have:
:$\map {\hat f} x = \inf \set {\map g x : g \ge f \text { and } g \text { is affine and continuous} }$ for each $x \in C$.
If there exists no continuous affine functional $g \ge f$, then $\map {\hat f} x = \infty$ for each $x \in C$.
Then, from Positive Infinity Function is Convex and Concave, $\hat f$ is conc... | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\R$.
Let $C \subseteq X$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Closed Set|closed]] [[Definition:Convex Set (Vector Space)|convex set]].
Let $f : C \to \R$ be a [[Definition:Function|function]].
Let ... | We have:
:$\map {\hat f} x = \inf \set {\map g x : g \ge f \text { and } g \text { is affine and continuous} }$ for each $x \in C$.
If there exists no [[Definition:Continuous Mapping|continuous]] [[Definition:Affine Functional|affine functional]] $g \ge f$, then $\map {\hat f} x = \infty$ for each $x \in C$.
Then, fr... | Concave Envelope is Concave Majorant of Function | https://proofwiki.org/wiki/Concave_Envelope_is_Concave_Majorant_of_Function | https://proofwiki.org/wiki/Concave_Envelope_is_Concave_Majorant_of_Function | [
"Concave Envelopes"
] | [
"Definition:Topological Vector Space",
"Definition:Non-Empty Set",
"Definition:Closed Set",
"Definition:Convex Set (Vector Space)",
"Definition:Function",
"Definition:Concave Envelope",
"Definition:Concave Real Function"
] | [
"Definition:Continuous Mapping",
"Definition:Affine Functional",
"Positive Infinity Function is Convex and Concave",
"Definition:Concave Real Function",
"Definition:Continuous Mapping",
"Definition:Affine Functional",
"Pointwise Infimum of Family of Concave Real Functions is Concave",
"Definition:Conc... |
proofwiki-23051 | Composite of Isomorphisms of Categories is Isomorphism | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be metacategories.
Let $F: \mathbf C \to \mathbf D$ and $G: \mathbf D \to \mathbf E$ be isomorphisms of categories.
Let $GF: \mathbf C \to \mathbf E$ be the composition of $G$ with $F$.
Let $F^{-1}: \mathbf D \to \mathbf C$ and $G^{-1}: \mathbf E \to \mathbf D$ be the inverse ... | We have:
{{begin-eqn}}
{{eqn | l = \paren{F^{-1} G^{-1} } \paren{G F}
| r = F^{-1} \paren{G^{-1} \paren{G F} }
| c = Composition of Functors is Associative
}}
{{eqn | r = F^{-1} \paren{\paren{G^{-1} G} F}
| c = Composition of Functors is Associative
}}
{{eqn | r = F^{-1} \paren{\operatorname{id}_{\mat... | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be [[Definition:Metacategory|metacategories]].
Let $F: \mathbf C \to \mathbf D$ and $G: \mathbf D \to \mathbf E$ be [[Definition:Isomorphism of Categories|isomorphisms of categories]].
Let $GF: \mathbf C \to \mathbf E$ be the [[Definition:Composition of Functors|composition ... | We have:
{{begin-eqn}}
{{eqn | l = \paren{F^{-1} G^{-1} } \paren{G F}
| r = F^{-1} \paren{G^{-1} \paren{G F} }
| c = [[Composition of Functors is Associative]]
}}
{{eqn | r = F^{-1} \paren{\paren{G^{-1} G} F}
| c = [[Composition of Functors is Associative]]
}}
{{eqn | r = F^{-1} \paren{\operatorname{i... | Composite of Isomorphisms of Categories is Isomorphism | https://proofwiki.org/wiki/Composite_of_Isomorphisms_of_Categories_is_Isomorphism | https://proofwiki.org/wiki/Composite_of_Isomorphisms_of_Categories_is_Isomorphism | [
"Isomorphisms of Categories"
] | [
"Definition:Metacategory",
"Definition:Isomorphism of Categories",
"Definition:Composition of Functors",
"Definition:Isomorphism of Categories/Inverse",
"Definition:Isomorphism of Categories",
"Definition:Isomorphism of Categories/Inverse",
"Definition:Isomorphism of Categories/Inverse"
] | [
"Composition of Functors is Associative",
"Composition of Functors is Associative",
"Composition of Functors is Associative",
"Composition of Functors is Associative",
"Definition:Isomorphism of Categories",
"Definition:Isomorphism of Categories/Inverse",
"Category:Isomorphisms of Categories"
] |
proofwiki-23052 | Convex Envelope is Convex Minorant of Function | Let $\struct {X, \tau}$ be a topological vector space over $\R$.
Let $C \subseteq X$ be a non-empty closed convex set.
Let $f : C \to \R$ be a function.
Let $\check f : C \to \overline \R$ be the convex envelope of $f$.
Then $\check f$ is convex and $\check f \le f$. | We have:
:$\map {\check f} x = \sup \set {\map g x : g \le f \text { and } g \text { is affine and continuous} }$ for each $x \in C$.
If there exists no continuous affine functional $g \le f$, then $\map {\check f} x = -\infty$ for each $x \in C$.
Then, from Negative Infinity Function is Convex and Concave, $\hat f$ is... | Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]] over $\R$.
Let $C \subseteq X$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Closed Set|closed]] [[Definition:Convex Set (Vector Space)|convex set]].
Let $f : C \to \R$ be a [[Definition:Function|function]].
Let ... | We have:
:$\map {\check f} x = \sup \set {\map g x : g \le f \text { and } g \text { is affine and continuous} }$ for each $x \in C$.
If there exists no [[Definition:Continuous Mapping|continuous]] [[Definition:Affine Functional|affine functional]] $g \le f$, then $\map {\check f} x = -\infty$ for each $x \in C$.
The... | Convex Envelope is Convex Minorant of Function | https://proofwiki.org/wiki/Convex_Envelope_is_Convex_Minorant_of_Function | https://proofwiki.org/wiki/Convex_Envelope_is_Convex_Minorant_of_Function | [
"Convex Envelopes"
] | [
"Definition:Topological Vector Space",
"Definition:Non-Empty Set",
"Definition:Closed Set",
"Definition:Convex Set (Vector Space)",
"Definition:Function",
"Definition:Convex Envelope",
"Definition:Convex Real Function"
] | [
"Definition:Continuous Mapping",
"Definition:Affine Functional",
"Negative Infinity Function is Convex and Concave",
"Definition:Convex Real Function",
"Definition:Continuous Mapping",
"Definition:Affine Functional",
"Pointwise Infimum of Family of Concave Real Functions is Concave",
"Definition:Conca... |
proofwiki-23053 | Characterization of Locally Convex Spaces | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a topological vector space.
Then $\tau$ is the standard topology of a locally convex space $\struct {X, \PP}$ {{iff}}:
:$\tau$ has a local basis $\BB_{ {\mathbf 0}_X}$ of ${\mathbf 0}_X$ consisting of absorbing, balanced, convex sets. | === Necessary Condition ===
Suppose that $\tau$ is standard topology of a locally convex space $\struct {X, \PP}$.
Let $U \in \tau$ be an open neighborhood of ${\mathbf 0}_X$.
From Open Sets in Standard Topology of Locally Convex Space, for each $y \in X$ there exists a finite set $F \subseteq \PP$ and $\epsilon > 0$ ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a [[Definition:Topological Vector Space|topological vector space]].
Then $\tau$ is the [[Definition:Locally Convex Space/Standard Topology|standard topology]] of a [[Definition:Locally Convex Space|locally convex space]] $\struct {X, \PP}$ {{iff}}:
:$\tau$ has ... | === Necessary Condition ===
Suppose that $\tau$ is [[Definition:Locally Convex Space/Standard Topology|standard topology]] of a [[Definition:Locally Convex Space|locally convex space]] $\struct {X, \PP}$.
Let $U \in \tau$ be an [[Definition:Open Neighborhood|open neighborhood]] of ${\mathbf 0}_X$.
From [[Open Sets ... | Characterization of Locally Convex Spaces | https://proofwiki.org/wiki/Characterization_of_Locally_Convex_Spaces | https://proofwiki.org/wiki/Characterization_of_Locally_Convex_Spaces | [
"Locally Convex Spaces",
"Characterization of Locally Convex Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Locally Convex Space",
"Definition:Local Basis",
"Definition:Absorbing Set",
"Definition:Balanced Set",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Locally Convex Space/Standard Topology",
"Definition:Locally Convex Space",
"Definition:Open Neighborhood",
"Open Sets in Standard Topology of Locally Convex Space",
"Definition:Finite Set",
"Definition:Open Set",
"Open Ball with respect to Seminorm is Convex, Balanced and Absorbing",
"Def... |
proofwiki-23054 | Cartesian Product of Convex Sets is Convex in Direct Product | Let $\GF \in \set {\R, \C}$.
Let $I$ be a set.
Let $\family {X_\alpha}_{\alpha \in I}$ be an $I$-indexed family of vector spaces over $\GF$.
Let:
:$\ds X = \prod_{\alpha \mathop \in I} X_\alpha$ be the direct product of $\family {X_\alpha}_{\alpha \in I}$.
For each $\alpha \in I$, let $C_\alpha \subseteq X_\alpha$ be ... | Let $x, y \in C$ and $t \in \closedint 0 1$.
Let $\alpha \in I$.
Then $\map x \alpha, \map y \alpha \in C_\alpha$.
We have:
:$\map {\paren {t x + \paren {1 - t} y} } \alpha = t \map x \alpha + \paren {1 - t} \map y \alpha \in C_\alpha$
since $C_\alpha$ is convex.
Hence:
:$\map {\paren {t x + \paren {1 - t} y} } \alpha ... | Let $\GF \in \set {\R, \C}$.
Let $I$ be a [[Definition:Set|set]].
Let $\family {X_\alpha}_{\alpha \in I}$ be an [[Definition:Indexed Family of Sets|$I$-indexed family]] of [[Definition:Vector Space|vector spaces]] over $\GF$.
Let:
:$\ds X = \prod_{\alpha \mathop \in I} X_\alpha$ be the [[Definition:Direct Product o... | Let $x, y \in C$ and $t \in \closedint 0 1$.
Let $\alpha \in I$.
Then $\map x \alpha, \map y \alpha \in C_\alpha$.
We have:
:$\map {\paren {t x + \paren {1 - t} y} } \alpha = t \map x \alpha + \paren {1 - t} \map y \alpha \in C_\alpha$
since $C_\alpha$ is [[Definition:Convex Set (Vector Space)|convex]].
Hence:
:$\m... | Cartesian Product of Convex Sets is Convex in Direct Product | https://proofwiki.org/wiki/Cartesian_Product_of_Convex_Sets_is_Convex_in_Direct_Product | https://proofwiki.org/wiki/Cartesian_Product_of_Convex_Sets_is_Convex_in_Direct_Product | [
"Direct Products of Vector Spaces",
"Direct Product of Vector Spaces",
"Direct Product of Vector Spaces",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Vector Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Non-Empty Set",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Category:Direct Product of Vector Spaces",
"Category:Convex Sets (Vector Spaces)"
] |
proofwiki-23055 | Direct Product of Locally Convex Spaces is Locally Convex | Let $\GF \in \set {\R, \C}$.
Let $I$ be a non-empty set.
For each $\alpha \in I$, let $\struct {X_\alpha, \PP_\alpha}$ be a locally convex space over $\GF$ with standard topology $\tau_\alpha$.
That is, $\struct {X_\alpha, \PP_\alpha}$ is a locally convex topological vector space.
Let:
:$\ds X = \prod_{\alpha \mathop \... | From Characterization of Locally Convex Spaces: Corollary, it is enough to establish that $\struct {X, \tau}$ has a local basis of ${\mathbf 0}_X$ consisting of convex sets.
Let $U$ be an open neighborhood of ${\mathbf 0}_X$ in $\struct {X, \tau}$.
By Natural Basis of Product Topology, there exists a finite set $F \sub... | Let $\GF \in \set {\R, \C}$.
Let $I$ be a [[Definition:Non-Empty Set|non-empty set]].
For each $\alpha \in I$, let $\struct {X_\alpha, \PP_\alpha}$ be a [[Definition:Locally Convex Space|locally convex space]] over $\GF$ with [[Definition:Locally Convex Space/Standard Topology|standard topology]] $\tau_\alpha$.
That... | From [[Characterization of Locally Convex Spaces/Corollary|Characterization of Locally Convex Spaces: Corollary]], it is enough to establish that $\struct {X, \tau}$ has a [[Definition:Local Basis|local basis]] of ${\mathbf 0}_X$ consisting of [[Definition:Convex Set (Vector Space)|convex sets]].
Let $U$ be an [[Defin... | Direct Product of Locally Convex Spaces is Locally Convex | https://proofwiki.org/wiki/Direct_Product_of_Locally_Convex_Spaces_is_Locally_Convex | https://proofwiki.org/wiki/Direct_Product_of_Locally_Convex_Spaces_is_Locally_Convex | [
"Locally Convex Spaces"
] | [
"Definition:Non-Empty Set",
"Definition:Locally Convex Space",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Locally Convex Topological Vector Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Product Topology",
"Definition:Locally Convex Space/Standard Topology",
"D... | [
"Characterization of Locally Convex Spaces/Corollary",
"Definition:Local Basis",
"Definition:Convex Set (Vector Space)",
"Definition:Open Neighborhood",
"Natural Basis of Product Topology",
"Definition:Finite Set",
"Definition:Locally Convex Topological Vector Space",
"Definition:Convex Set (Vector Sp... |
proofwiki-23056 | Existence of Continuous Affine Minorant of Lower Semicontinuous Proper Convex Function with Prescribed Point Evaluation | Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\R$.
Let $f : X \to \overline \R$ be a lower semicontinuous proper convex function.
Let $\Dom f$ be the effective domain of $f$.
Let $x \in \Dom f$.
Let $\alpha \in \R$ such that:
:$\alpha < \map f x$
Then there exists a continuous affine functional $g : ... | Let $\struct {X \times \R}^\ast$ be the topological dual of the direct product $X \times \R$.
Let $\map {\operatorname {epi} } f \subseteq X \times \R$ be the epigraph of $f$.
Since $\map f x < \infty$, there exists $\beta \in \R$ such that $\map f x < \beta$.
Hence $\tuple {x, \beta} \in \map {\operatorname {epi} } f... | Let $\struct {X, \PP}$ be a [[Definition:Hausdorff Locally Convex Space|Hausdorff locally convex space]] over $\R$.
Let $f : X \to \overline \R$ be a [[Definition:Lower Semicontinuous|lower semicontinuous]] [[Definition:Proper Convex Real Function|proper convex function]].
Let $\Dom f$ be the [[Definition:Effective D... | Let $\struct {X \times \R}^\ast$ be the [[Definition:Topological Dual Space|topological dual]] of the [[Definition:Direct Product of Vector Spaces|direct product]] $X \times \R$.
Let $\map {\operatorname {epi} } f \subseteq X \times \R$ be the [[Definition:Epigraph|epigraph]] of $f$.
Since $\map f x < \infty$, there... | Existence of Continuous Affine Minorant of Lower Semicontinuous Proper Convex Function with Prescribed Point Evaluation | https://proofwiki.org/wiki/Existence_of_Continuous_Affine_Minorant_of_Lower_Semicontinuous_Proper_Convex_Function_with_Prescribed_Point_Evaluation | https://proofwiki.org/wiki/Existence_of_Continuous_Affine_Minorant_of_Lower_Semicontinuous_Proper_Convex_Function_with_Prescribed_Point_Evaluation | [
"Lower Semicontinuity",
"Convex Real Functions"
] | [
"Definition:Locally Convex Space/Hausdorff",
"Definition:Lower Semicontinuous",
"Definition:Proper Convex Real Function",
"Definition:Effective Domain of Convex Real Function",
"Definition:Continuous Mapping",
"Definition:Affine Functional"
] | [
"Definition:Topological Dual Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Epigraph",
"Definition:Lower Semicontinuous",
"Definition:Closed Set",
"Characterization of Lower Semicontinuity",
"Definition:Convex Real Function/Vector Space",
"Definition:Convex Set (Vector Space)",
"F... |
proofwiki-23057 | Convex Envelope of Lower Semicontinuous Proper Convex Function is Equal to Original Function | Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\R$.
Let $f : X \to \overline \R$ be a lower semicontinuous proper convex function.
Let $\check f$ be the convex envelope of $f$.
Then $\check f = f$. | From Convex Envelope is Convex Minorant of Function, we have $\check f \le f$.
Let $\Dom f$ be the effective domain of $f$.
Let $\map {\operatorname {epi} } f$ be the epigraph of $f$.
Let $x \in X$.
Then $-\infty < \map f x$.
We argue that it is sufficient to show that for all $\alpha \in \R$ with $\alpha < \map f x$,... | Let $\struct {X, \PP}$ be a [[Definition:Hausdorff Locally Convex Space|Hausdorff locally convex space]] over $\R$.
Let $f : X \to \overline \R$ be a [[Definition:Lower Semicontinuous|lower semicontinuous]] [[Definition:Proper Convex Real Function|proper convex function]].
Let $\check f$ be the [[Definition:Convex En... | From [[Convex Envelope is Convex Minorant of Function]], we have $\check f \le f$.
Let $\Dom f$ be the [[Definition:Effective Domain of Convex Real Function|effective domain]] of $f$.
Let $\map {\operatorname {epi} } f$ be the [[Definition:Epigraph|epigraph]] of $f$.
Let $x \in X$.
Then $-\infty < \map f x$.
We ... | Convex Envelope of Lower Semicontinuous Proper Convex Function is Equal to Original Function | https://proofwiki.org/wiki/Convex_Envelope_of_Lower_Semicontinuous_Proper_Convex_Function_is_Equal_to_Original_Function | https://proofwiki.org/wiki/Convex_Envelope_of_Lower_Semicontinuous_Proper_Convex_Function_is_Equal_to_Original_Function | [
"Convex Envelopes"
] | [
"Definition:Locally Convex Space/Hausdorff",
"Definition:Lower Semicontinuous",
"Definition:Proper Convex Real Function",
"Definition:Convex Envelope"
] | [
"Convex Envelope is Convex Minorant of Function",
"Definition:Effective Domain of Convex Real Function",
"Definition:Epigraph",
"Definition:Continuous Mapping",
"Definition:Affine Functional",
"Definition:Continuous Mapping",
"Definition:Affine Functional",
"Existence of Continuous Affine Minorant of ... |
proofwiki-23058 | Characterization of Subgradient in terms of Maximizer of Linear Functional | Let $\innerprod X {X'}_X$ be a dual system over $\R$.
Let $f : X \to \overline \R$ be a proper convex function.
Let $\Dom f$ be the effective domain of $f$.
Define the canonical duality $\innerprod \cdot \cdot_{X \times \R} : \paren {X \times \R} \times \paren {X' \times \R} \to \R$ by:
:$\innerprod {\tuple {x, \alpha... | We have that $x'$ is a subgradient for $f$ {{iff}}:
:$\map f y \ge \map f x + \innerprod {y - x} {x'}$ for each $y \in \Dom f$.
We can write this as:
:$\innerprod x {x'}_X - \map f x \ge \innerprod y {x'}_X - \map f y$ for each $y \in \Dom f$. | Let $\innerprod X {X'}_X$ be a [[Definition:Dual System|dual system]] over $\R$.
Let $f : X \to \overline \R$ be a [[Definition:Proper Convex Real Function|proper convex function]].
Let $\Dom f$ be the [[Definition:Effective Domain of Convex Real Function|effective domain]] of $f$.
Define the [[Definition:Canonical... | We have that $x'$ is a [[Definition:Subgradient|subgradient]] for $f$ {{iff}}:
:$\map f y \ge \map f x + \innerprod {y - x} {x'}$ for each $y \in \Dom f$.
We can write this as:
:$\innerprod x {x'}_X - \map f x \ge \innerprod y {x'}_X - \map f y$ for each $y \in \Dom f$. | Characterization of Subgradient in terms of Maximizer of Linear Functional | https://proofwiki.org/wiki/Characterization_of_Subgradient_in_terms_of_Maximizer_of_Linear_Functional | https://proofwiki.org/wiki/Characterization_of_Subgradient_in_terms_of_Maximizer_of_Linear_Functional | [
"Subgradients"
] | [
"Definition:Dual System",
"Definition:Proper Convex Real Function",
"Definition:Effective Domain of Convex Real Function",
"Definition:Canonical Duality of Product of Topological Vector Space with its Scalar Field and its Dual",
"Definition:Linear Functional",
"Definition:Epigraph",
"Definition:Subgradi... | [
"Definition:Subgradient",
"Definition:Subgradient",
"Definition:Subgradient"
] |
proofwiki-23059 | Distance Travelled during Time Interval | Let $P$ be a particle in space.
Let $\map {\mathbf r} t$ denote the displacement from $\map {\mathbf r} 0$ of $P$ at time $t$ where $0 \le t \le T$ for some instant $T$.
Then the total distance travelled by $P$ between $t = 0$ and $t = T$ is given by:
:$\map s T = \ds \int_0^T \size {\dfrac {\d \mathbf r} {\d t} } \rd ... | At a given time $t$, the particle $P$ has an instantaneous velocity
:$\map {\mathbf v} t = \dfrac {\d \mathbf r} {\d t}$
During the infinitesimal time interval $\d t$, $P$ moves a distance of:
{{begin-eqn}}
{{eqn | l = \d s
| r = \size {\map {\mathbf v} t \d t}
| c = magnitude of displacement is distance
}}... | Let $P$ be a [[Definition:Particle|particle]] in [[Definition:Ordinary Space|space]].
Let $\map {\mathbf r} t$ denote the [[Definition:Displacement|displacement]] from $\map {\mathbf r} 0$ of $P$ at [[Definition:Time|time]] $t$ where $0 \le t \le T$ for some [[Definition:Time Instant|instant]] $T$.
Then the total [[... | At a given [[Definition:Time|time]] $t$, the [[Definition:Particle|particle]] $P$ has an [[Definition:Instantaneous Velocity|instantaneous velocity]]
:$\map {\mathbf v} t = \dfrac {\d \mathbf r} {\d t}$
During the [[Definition:Infinitesimal|infinitesimal]] [[Definition:Time Interval|time interval]] $\d t$, $P$ moves a... | Distance Travelled during Time Interval | https://proofwiki.org/wiki/Distance_Travelled_during_Time_Interval | https://proofwiki.org/wiki/Distance_Travelled_during_Time_Interval | [
"Distance (Geometry)",
"Velocity",
"Speed"
] | [
"Definition:Particle",
"Definition:Ordinary Space",
"Definition:Displacement",
"Definition:Time",
"Definition:Instant of Time",
"Definition:Distance (Geometry)",
"Definition:Magnitude"
] | [
"Definition:Time",
"Definition:Particle",
"Definition:Instantaneous/Velocity",
"Definition:Infinitesimal",
"Definition:Time/Length",
"Definition:Distance (Geometry)",
"Absolute Value Function is Completely Multiplicative",
"Definition:Positive/Real Number",
"Definition:Distance (Geometry)"
] |
proofwiki-23060 | Perpendicular is Shortest Straight Line from Point to Plane | Let $\PP$ be a plane.
Let $C$ be a point which is not on $\PP$.
Let $D$ be a point on $\PP$ such that $CD$ is perpendicular to $\PP$.
Then the length of $CD$ is less than the length of all other line segments that can be drawn from $C$ to $\PP$. | Let $E$ on $\PP$ such that $E$ is different from $D$.
Then $CDE$ forms a right triangle where $CE$ is the hypotenuse.
By Pythagoras's Theorem:
:$CD^2 + DE^2 = CE^2$
and so $CD < CE$.
{{qed}} | Let $\PP$ be a [[Definition:Plane|plane]].
Let $C$ be a [[Definition:Point|point]] which is not on $\PP$.
Let $D$ be a [[Definition:Point|point]] on $\PP$ such that $CD$ is [[Definition:Line Perpendicular to Plane|perpendicular]] to $\PP$.
Then the [[Definition:Length of Line|length]] of $CD$ is less than the [[Def... | Let $E$ on $\PP$ such that $E$ is different from $D$.
Then $CDE$ forms a [[Definition:Right Triangle|right triangle]] where $CE$ is the [[Definition:Hypotenuse|hypotenuse]].
By [[Pythagoras's Theorem]]:
:$CD^2 + DE^2 = CE^2$
and so $CD < CE$.
{{qed}} | Perpendicular is Shortest Straight Line from Point to Plane | https://proofwiki.org/wiki/Perpendicular_is_Shortest_Straight_Line_from_Point_to_Plane | https://proofwiki.org/wiki/Perpendicular_is_Shortest_Straight_Line_from_Point_to_Plane | [
"Perpendicular Distance between Point and Plane"
] | [
"Definition:Plane Surface",
"Definition:Point",
"Definition:Point",
"Definition:Right Angle/Perpendicular/Plane",
"Definition:Linear Measure/Length",
"Definition:Linear Measure/Length",
"Definition:Line/Segment"
] | [
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Pythagoras's Theorem"
] |
proofwiki-23061 | Vector Equation of Straight Line in Space/Formulation 1 | Let $\mathbf a$ and $\mathbf b$ denote the position vectors of two points in space.
Let $L$ be a straight line in space passing through $\mathbf a$ which is parallel to $\mathbf b$.
Let $\mathbf r$ be the position vector of an arbitrary point on $L$.
Then:
:$\mathbf r = \mathbf a + t \mathbf b$
for some real number $t$... | :300px
Let $a$ and $b$ be points as given, with their position vectors $\mathbf a$ and $\mathbf b$ respectively.
Let $P$ be an arbitrary point on the straight line $L$ passing through $\mathbf a$ which is parallel to $\mathbf b$.
By the parallel postulate, $L$ exists and is unique.
Let $\mathbf r$ be the position vecto... | Let $\mathbf a$ and $\mathbf b$ denote the [[Definition:Position Vector|position vectors]] of two [[Definition:Point|points]] in [[Definition:Ordinary Space|space]].
Let $L$ be a [[Definition:Straight Line|straight line]] in [[Definition:Ordinary Space|space]] passing through $\mathbf a$ which is [[Definition:Parallel... | :[[File:Vector-equation-of-straight-line.png|300px]]
Let $a$ and $b$ be [[Definition:Point|points]] as given, with their [[Definition:Position Vector|position vectors]] $\mathbf a$ and $\mathbf b$ respectively.
Let $P$ be an arbitrary [[Definition:Point|point]] on the [[Definition:Straight Line|straight line]] $L$ p... | Vector Equation of Straight Line in Space/Formulation 1 | https://proofwiki.org/wiki/Vector_Equation_of_Straight_Line_in_Space/Formulation_1 | https://proofwiki.org/wiki/Vector_Equation_of_Straight_Line_in_Space/Formulation_1 | [
"Vector Equation of Straight Line in Space",
"Equations of Straight Lines in Space",
"Vectors"
] | [
"Definition:Position Vector",
"Definition:Point",
"Definition:Ordinary Space",
"Definition:Line/Straight Line",
"Definition:Ordinary Space",
"Definition:Parallel (Geometry)/Lines",
"Definition:Position Vector",
"Definition:Point",
"Definition:Real Number",
"Definition:Positive/Real Number",
"Def... | [
"File:Vector-equation-of-straight-line.png",
"Definition:Point",
"Definition:Position Vector",
"Definition:Point",
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Axiom:Parallel Postulate",
"Definition:Unique",
"Definition:Position Vector",
"Definition:Parallel (Geometry)... |
proofwiki-23062 | Vector Equation of Straight Line in Space/Formulation 2 | Let $\mathbf a$ and $\mathbf b$ denote the position vectors of two points in space.
Let $L$ be a straight line in space passing through $\mathbf a$ and $\mathbf b$.
Let $\mathbf r$ be the position vector of an arbitrary point on $L$.
Then:
:$\mathbf r = \mathbf a + t \paren {\mathbf b - \mathbf a}$
for some real number... | From {{Formulation|Vector Equation of Straight Line in Space|1}}:
{{:Vector Equation of Straight Line in Space/Formulation 1}}
{{qed|lemma}}
:420px
Let $a$ and $b$ be points as given, with their position vectors $\mathbf a$ and $\mathbf b$ respectively.
Let $P$ be an arbitrary point on the straight line $L$ passing thr... | Let $\mathbf a$ and $\mathbf b$ denote the [[Definition:Position Vector|position vectors]] of two [[Definition:Point|points]] in [[Definition:Ordinary Space|space]].
Let $L$ be a [[Definition:Straight Line|straight line]] in [[Definition:Ordinary Space|space]] passing through $\mathbf a$ and $\mathbf b$.
Let $\mathbf... | From {{Formulation|Vector Equation of Straight Line in Space|1}}:
{{:Vector Equation of Straight Line in Space/Formulation 1}}
{{qed|lemma}}
:[[File:Vector-equation-of-straight-line-formulation-2.png|420px]]
Let $a$ and $b$ be [[Definition:Point|points]] as given, with their [[Definition:Position Vector|position ve... | Vector Equation of Straight Line in Space/Formulation 2 | https://proofwiki.org/wiki/Vector_Equation_of_Straight_Line_in_Space/Formulation_2 | https://proofwiki.org/wiki/Vector_Equation_of_Straight_Line_in_Space/Formulation_2 | [
"Vector Equation of Straight Line in Space",
"Equations of Straight Lines in Space",
"Vectors"
] | [
"Definition:Position Vector",
"Definition:Point",
"Definition:Ordinary Space",
"Definition:Line/Straight Line",
"Definition:Ordinary Space",
"Definition:Position Vector",
"Definition:Point",
"Definition:Real Number",
"Definition:Positive/Real Number",
"Definition:Negative/Real Number"
] | [
"File:Vector-equation-of-straight-line-formulation-2.png",
"Definition:Point",
"Definition:Position Vector",
"Definition:Point",
"Definition:Line/Straight Line",
"Definition:Vector Subtraction",
"Definition:Vector Quantity",
"Definition:Parallel (Geometry)/Lines"
] |
proofwiki-23063 | Slice Category is Isomorphic to Identity Functor Over Object Comma Category/Lemma 1 | :$I = J_2 \circ I_1$ | For each object $g$ in $\mathbf C / C$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J_2 \circ I_1} g
| r = \map {J_2} {I_1 g}
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J_2 \tuple{\operatorname{dom} g, *, g}
| c = Definition of $I_1$
}}
{{eqn | r = \tuple{\operatorname{dom} g, g}
| c = Defin... | :$I = J_2 \circ I_1$ | For each [[Definition:Object (Category Theory)|object]] $g$ in $\mathbf C / C$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J_2 \circ I_1} g
| r = \map {J_2} {I_1 g}
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J_2 \tuple{\operatorname{dom} g, *, g}
| c = Definition of $I_1$
}}
{{eqn | r = \tuple{\o... | Slice Category is Isomorphic to Identity Functor Over Object Comma Category/Lemma 1 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Identity_Functor_Over_Object_Comma_Category/Lemma_1 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Identity_Functor_Over_Object_Comma_Category/Lemma_1 | [
"Slice Category is Isomorphic to Identity Functor Over Object Comma Category"
] | [] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Slice Category is Isomorphic to Identity Functor Over Object Comma Category"
] |
proofwiki-23064 | Slice Category is Isomorphic to Identity Functor Over Object Comma Category/Lemma 2 | :$J = J_1 \circ I_2$ | For each object $\tuple{E, g}$ in $\paren{\operatorname{id}_{\mathbf C} \downarrow C }$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J_1 \circ I_2} \tuple{E, g}
| r = \map {J_1} {I_2 \tuple{E, g} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J_1 \tuple{E, *, g}
| c = Definition of $I_2$
}}
{{eqn | r... | :$J = J_1 \circ I_2$ | For each [[Definition:Object (Category Theory)|object]] $\tuple{E, g}$ in $\paren{\operatorname{id}_{\mathbf C} \downarrow C }$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J_1 \circ I_2} \tuple{E, g}
| r = \map {J_1} {I_2 \tuple{E, g} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J_1 \tuple{E, *, g}
... | Slice Category is Isomorphic to Identity Functor Over Object Comma Category/Lemma 2 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Identity_Functor_Over_Object_Comma_Category/Lemma_2 | https://proofwiki.org/wiki/Slice_Category_is_Isomorphic_to_Identity_Functor_Over_Object_Comma_Category/Lemma_2 | [
"Slice Category is Isomorphic to Identity Functor Over Object Comma Category"
] | [] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Slice Category is Isomorphic to Identity Functor Over Object Comma Category"
] |
proofwiki-23065 | Banach Space-Valued Simple Function takes Finitely Many Values | Let $\GF \in \set {\R, \C}$.
Let $I$ be a real interval.
Let $X$ be a Banach space over $\GF$.
Let $f : I \to X$ be a simple function.
Then $\Img f$ is a finite set. | Since $f$ is a simple function, there exists:
:Lebesgue measurable subsets $\Omega_1, \ldots, \Omega_n$ of $I$ with finite Lebesgue measure
:$x_1, \ldots, x_n \in X$
such that:
:$\ds \map f t = \sum_{r \mathop = 1}^n x_r \map {\chi_{\Omega_r} } t$
for each $t \in I$.
Note that $\map {\chi_{\Omega_r} } t \in \set {0,... | Let $\GF \in \set {\R, \C}$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $X$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $f : I \to X$ be a [[Definition:Simple Function/Banach Space|simple function]].
Then $\Img f$ is a [[Definition:Finite Set|finite set]]. | Since $f$ is a [[Definition:Simple Function/Banach Space|simple function]], there exists:
:[[Definition:Measurable Set/Subsets of Real Space|Lebesgue measurable subsets]] $\Omega_1, \ldots, \Omega_n$ of $I$ with [[Definition:Finite Extended Real Number|finite]] [[Definition:Lebesgue Measure|Lebesgue measure]]
:$x_1, \... | Banach Space-Valued Simple Function takes Finitely Many Values | https://proofwiki.org/wiki/Banach_Space-Valued_Simple_Function_takes_Finitely_Many_Values | https://proofwiki.org/wiki/Banach_Space-Valued_Simple_Function_takes_Finitely_Many_Values | [
"Simple Functions",
"Banach Spaces"
] | [
"Definition:Real Interval",
"Definition:Banach Space",
"Definition:Simple Function/Banach Space",
"Definition:Finite Set"
] | [
"Definition:Simple Function/Banach Space",
"Definition:Measurable Set/Subsets of Real Space",
"Definition:Finite Extended Real Number",
"Definition:Lebesgue Measure",
"Cardinality of Image of Set not greater than Cardinality of Set",
"Definition:Finite Set",
"Subset of Finite Set is Finite",
"Definiti... |
proofwiki-23066 | Banach Space-Valued Simple Function has Standard Representation | Let $\GF \in \set {\R, \C}$.
Let $I$ be a real interval.
Let $X$ be a Banach space over $\GF$.
Let $f : I \to X$ be a simple function.
Then $f$ is a simple function {{iff}} there exists:
:disjoint Lebesgue measurable sets $\Omega_1, \ldots, \Omega_n \subseteq I$ of finite Lebesgue measure
:$x_1, \ldots, x_n \in X$
su... | From the definition of a simple function, there exists:
:real intervals $\Omega_1', \ldots, \Omega_n' \subseteq I$ of finite Lebesgue measure
:$x_1, \ldots, x_n \in X$
such that:
:$\ds \map f t = \sum_{r \mathop = 1}^n x_r \map {\chi_{\Omega_r} } t$ for each $t \in I$.
{{WLOG}} suppose that:
:$\ds \bigcup_{r \mathop = ... | Let $\GF \in \set {\R, \C}$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $X$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $f : I \to X$ be a [[Definition:Simple Function|simple function]].
Then $f$ is a [[Definition:Simple Function/Banach Space|simple function]] {{iff}} there ex... | From the definition of a [[Definition:Simple Function|simple function]], there exists:
:[[Definition:Real Interval|real intervals]] $\Omega_1', \ldots, \Omega_n' \subseteq I$ of [[Definition:Finite Extended Real Number|finite]] [[Definition:Lebesgue Measure|Lebesgue measure]]
:$x_1, \ldots, x_n \in X$
such that:
:$\ds ... | Banach Space-Valued Simple Function has Standard Representation | https://proofwiki.org/wiki/Banach_Space-Valued_Simple_Function_has_Standard_Representation | https://proofwiki.org/wiki/Banach_Space-Valued_Simple_Function_has_Standard_Representation | [
"Banach Spaces",
"Simple Functions"
] | [
"Definition:Real Interval",
"Definition:Banach Space",
"Definition:Simple Function",
"Definition:Simple Function/Banach Space",
"Definition:Disjoint Sets",
"Definition:Measurable Set",
"Definition:Finite Extended Real Number",
"Definition:Lebesgue Measure"
] | [
"Definition:Simple Function",
"Definition:Real Interval",
"Definition:Finite Extended Real Number",
"Definition:Lebesgue Measure",
"Banach Space-Valued Simple Function takes Finitely Many Values",
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Disjoint Sets... |
proofwiki-23067 | Sum of Banach Space-Valued Simple Functions is Simple Function | Let $\GF \in \set {\R, \C}$.
Let $I$ be a real interval.
Let $X$ be a Banach space over $\GF$.
Let $f, g : I \to X$ be a simple function.
Then $f + g$ is simple. | From the definition of a simple function, there exists:
:Lebesgue measurable subsets $\II_1, \ldots, \II_n \subseteq I$ of finite Lebesgue measure
:$x_1, \ldots, x_n \in X$
such that:
:$\ds \map f t = \sum_{r \mathop = 1}^n x_r \map {\chi_{\II_r} } t$ for each $t \in I$.
From the definition of a simple function, there ... | Let $\GF \in \set {\R, \C}$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $X$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $f, g : I \to X$ be a [[Definition:Simple Function/Banach Space|simple function]].
Then $f + g$ is [[Definition:Simple Function/Banach Space|simple]]. | From the definition of a [[Definition:Simple Function|simple function]], there exists:
:[[Definition:Measurable Set/Subsets of Real Space|Lebesgue measurable subsets]] $\II_1, \ldots, \II_n \subseteq I$ of [[Definition:Finite Extended Real Number|finite]] [[Definition:Lebesgue Measure|Lebesgue measure]]
:$x_1, \ldots, ... | Sum of Banach Space-Valued Simple Functions is Simple Function | https://proofwiki.org/wiki/Sum_of_Banach_Space-Valued_Simple_Functions_is_Simple_Function | https://proofwiki.org/wiki/Sum_of_Banach_Space-Valued_Simple_Functions_is_Simple_Function | [
"Simple Functions",
"Banach Spaces"
] | [
"Definition:Real Interval",
"Definition:Banach Space",
"Definition:Simple Function/Banach Space",
"Definition:Simple Function/Banach Space"
] | [
"Definition:Simple Function",
"Definition:Measurable Set/Subsets of Real Space",
"Definition:Finite Extended Real Number",
"Definition:Lebesgue Measure",
"Definition:Simple Function",
"Definition:Measurable Set/Subsets of Real Space",
"Definition:Finite Extended Real Number",
"Definition:Lebesgue Meas... |
proofwiki-23068 | Sum of Bochner Measurable Functions is Bochner Measurable | Let $\GF \in \set {\R, \C}$.
Let $I$ be a real interval.
Let $X$ be a Banach space over $\GF$.
Let $f, g : I \to X$ be a Bochner measurable function.
Then $f + g$ is Bochner measurable. | Since $f$ is Bochner measurable, there exists a sequence of simple functions $\sequence {\phi_n}_{n \mathop \in \N}$ such that:
:$\ds \map f t = \lim_{n \mathop \to \infty} \map {\phi_n} t$ for almost all $t \in I$.
Since $g$ is Bochner measurable, there exists a sequence of simple functions $\sequence {\psi_n}_{n \mat... | Let $\GF \in \set {\R, \C}$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $X$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $f, g : I \to X$ be a [[Definition:Bochner Measurable Function|Bochner measurable function]].
Then $f + g$ is [[Definition:Bochner Measurable Function|Bochne... | Since $f$ is [[Definition:Bochner Measurable Function|Bochner measurable]], there exists a [[Definition:Sequence|sequence]] of [[Definition:Simple Function|simple functions]] $\sequence {\phi_n}_{n \mathop \in \N}$ such that:
:$\ds \map f t = \lim_{n \mathop \to \infty} \map {\phi_n} t$ for [[Definition:Almost All|almo... | Sum of Bochner Measurable Functions is Bochner Measurable | https://proofwiki.org/wiki/Sum_of_Bochner_Measurable_Functions_is_Bochner_Measurable | https://proofwiki.org/wiki/Sum_of_Bochner_Measurable_Functions_is_Bochner_Measurable | [
"Bochner Measurable Functions"
] | [
"Definition:Real Interval",
"Definition:Banach Space",
"Definition:Bochner Measurable Function",
"Definition:Bochner Measurable Function"
] | [
"Definition:Bochner Measurable Function",
"Definition:Sequence",
"Definition:Simple Function",
"Definition:Almost All",
"Definition:Bochner Measurable Function",
"Definition:Sequence",
"Definition:Simple Function",
"Definition:Almost All",
"Definition:Measurable Set/Subsets of Real Space",
"Defini... |
proofwiki-23069 | Left Composition of Banach Space-Valued Simple Function with Function Fixing Origin is Simple Function | Let $\GF \in \set {\R, \C}$.
Let $I$ be a real interval.
Let $X$ and $Y$ be Banach spaces over $\GF$.
Let $f : I \to X$ be a simple function.
Let $g : X \to Y$ be a function such that $\map g { {\mathbf 0}_X} = {\mathbf 0}_Y$.
Then $g \circ f$ is a simple function. | From Banach Space-Valued Simple Function has Standard Representation:
:disjoint Lebesgue measurable sets $\Omega_1, \ldots, \Omega_n \subseteq I$ of finite Lebesgue measure
:$x_1, \ldots, x_n \in X$
such that:
:$\ds \map f t = \sum_{r \mathop = 1}^n x_r \map {\chi_{\Omega_r} } t$ for each $t \in I$.
That is:
:$\map f t... | Let $\GF \in \set {\R, \C}$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $X$ and $Y$ be [[Definition:Banach Space|Banach spaces]] over $\GF$.
Let $f : I \to X$ be a [[Definition:Simple Function/Banach Space|simple function]].
Let $g : X \to Y$ be a [[Definition:Function|function]] such that $\map... | From [[Banach Space-Valued Simple Function has Standard Representation]]:
:[[Definition:Disjoint Sets|disjoint]] [[Definition:Measurable Set|Lebesgue measurable sets]] $\Omega_1, \ldots, \Omega_n \subseteq I$ of [[Definition:Finite Extended Real Number|finite]] [[Definition:Lebesgue Measure|Lebesgue measure]]
:$x_1, \l... | Left Composition of Banach Space-Valued Simple Function with Function Fixing Origin is Simple Function | https://proofwiki.org/wiki/Left_Composition_of_Banach_Space-Valued_Simple_Function_with_Function_Fixing_Origin_is_Simple_Function | https://proofwiki.org/wiki/Left_Composition_of_Banach_Space-Valued_Simple_Function_with_Function_Fixing_Origin_is_Simple_Function | [
"Simple Functions",
"Banach Spaces"
] | [
"Definition:Real Interval",
"Definition:Banach Space",
"Definition:Simple Function/Banach Space",
"Definition:Function",
"Definition:Simple Function/Banach Space"
] | [
"Banach Space-Valued Simple Function has Standard Representation",
"Definition:Disjoint Sets",
"Definition:Measurable Set",
"Definition:Finite Extended Real Number",
"Definition:Lebesgue Measure",
"Definition:Simple Function/Banach Space",
"Category:Simple Functions",
"Category:Banach Spaces"
] |
proofwiki-23070 | Subset of Set with Zero Outer Measure has Zero Outer Measure | Let $X$ be a set.
Let $\mu^\ast$ be an outer measure on $X$.
Let $B \subseteq X$ be such that $\map {\mu^\ast} B = 0$.
Let $A \subseteq B$.
Then $\map {\mu^\ast} A = 0$. | From condition $(2)$ in the definition of an outer measure, we have:
:$\map {\mu^\ast} A \le \map {\mu^\ast} B = 0$
Since $0 \le \map {\mu^\ast} A$, we have $\map {\mu^\ast} A = 0$.
{{qed}}
Category:Outer Measures
d09rwljzoak4xkbjrwx4mjr1ro2cfuz | Let $X$ be a [[Definition:Set|set]].
Let $\mu^\ast$ be an [[Definition:Outer Measure|outer measure]] on $X$.
Let $B \subseteq X$ be such that $\map {\mu^\ast} B = 0$.
Let $A \subseteq B$.
Then $\map {\mu^\ast} A = 0$. | From condition $(2)$ in the definition of an [[Definition:Outer Measure|outer measure]], we have:
:$\map {\mu^\ast} A \le \map {\mu^\ast} B = 0$
Since $0 \le \map {\mu^\ast} A$, we have $\map {\mu^\ast} A = 0$.
{{qed}}
[[Category:Outer Measures]]
d09rwljzoak4xkbjrwx4mjr1ro2cfuz | Subset of Set with Zero Outer Measure has Zero Outer Measure | https://proofwiki.org/wiki/Subset_of_Set_with_Zero_Outer_Measure_has_Zero_Outer_Measure | https://proofwiki.org/wiki/Subset_of_Set_with_Zero_Outer_Measure_has_Zero_Outer_Measure | [
"Outer Measures"
] | [
"Definition:Set",
"Definition:Outer Measure"
] | [
"Definition:Outer Measure",
"Category:Outer Measures"
] |
proofwiki-23071 | Measure Space arising from Outer Measure is Complete | Let $X$ be a set.
Let $\mu^\ast$ be an outer measure on $X$.
Let $\map {\mathfrak M} {\mu^*}$ be the collection of $\mu^*$-measurable sets.
Let $\mu$ be the restriction of $\mu^*$ to $\map {\mathfrak M} {\mu^*}$.
Then $\struct {X, \map {\mathfrak M} {\mu^\ast}, \mu}$ is a complete measure space. | From Measure Space from Outer Measure, $\struct {X, \map {\mathfrak M} {\mu^\ast}, \mu}$ is a measure space.
Let $A \subseteq X$ be such that $\map {\mu^\ast} A = 0$.
We aim to show that $A \in \map {\mathfrak M} {\mu^\ast}$.
Let $S \subseteq X$.
We want to establish that:
:$\map {\mu^\ast} A = \map {\mu^\ast} {A \cap ... | Let $X$ be a [[Definition:Set|set]].
Let $\mu^\ast$ be an [[Definition:Outer Measure|outer measure]] on $X$.
Let $\map {\mathfrak M} {\mu^*}$ be the collection of [[Definition:Measurable Set|$\mu^*$-measurable sets]].
Let $\mu$ be the [[Definition:Restriction of Mapping|restriction]] of $\mu^*$ to $\map {\mathfrak M... | From [[Measure Space from Outer Measure]], $\struct {X, \map {\mathfrak M} {\mu^\ast}, \mu}$ is a [[Definition:Measure Space|measure space]].
Let $A \subseteq X$ be such that $\map {\mu^\ast} A = 0$.
We aim to show that $A \in \map {\mathfrak M} {\mu^\ast}$.
Let $S \subseteq X$.
We want to establish that:
:$\map {\... | Measure Space arising from Outer Measure is Complete | https://proofwiki.org/wiki/Measure_Space_arising_from_Outer_Measure_is_Complete | https://proofwiki.org/wiki/Measure_Space_arising_from_Outer_Measure_is_Complete | [
"Complete Measure Spaces"
] | [
"Definition:Set",
"Definition:Outer Measure",
"Definition:Measurable Set",
"Definition:Restriction/Mapping",
"Definition:Complete Measure Space"
] | [
"Measure Space from Outer Measure",
"Definition:Measure Space",
"Intersection is Subset",
"Subset of Set with Zero Outer Measure has Zero Outer Measure",
"Definition:Complete Measure Space",
"Category:Complete Measure Spaces"
] |
proofwiki-23072 | Lebesgue Measure Space is Complete | Let $\map {\operatorname {Leb} } {\R^n}$ be the Lebesgue $\sigma$-algebra on $\R^n$.
Let $\lambda$ be the Lebesgue measure on $\R^n$.
Then $\struct {\R^n, \map {\operatorname {Leb} } {\R^n}, \lambda}$ is a complete measure space. | By definition, $\struct {\R^n, \map {\operatorname {Leb} } {\R^n}, \lambda}$ is the measure space that arises from the Lebesgue outer measure as in Measure Space from Outer Measure.
From Measure Space arising from Outer Measure is Complete, $\struct {\R^n, \map {\operatorname {Leb} } {\R^n}, \lambda}$ is a complete mea... | Let $\map {\operatorname {Leb} } {\R^n}$ be the [[Definition:Lebesgue Sigma-Algebra|Lebesgue $\sigma$-algebra]] on $\R^n$.
Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] on $\R^n$.
Then $\struct {\R^n, \map {\operatorname {Leb} } {\R^n}, \lambda}$ is a [[Definition:Complete Measure Space|comp... | By definition, $\struct {\R^n, \map {\operatorname {Leb} } {\R^n}, \lambda}$ is the [[Definition:Measure Space|measure space]] that arises from the [[Definition:Lebesgue Outer Measure|Lebesgue outer measure]] as in [[Measure Space from Outer Measure]].
From [[Measure Space arising from Outer Measure is Complete]], $\s... | Lebesgue Measure Space is Complete | https://proofwiki.org/wiki/Lebesgue_Measure_Space_is_Complete | https://proofwiki.org/wiki/Lebesgue_Measure_Space_is_Complete | [
"Lebesgue Measure"
] | [
"Definition:Lebesgue Sigma-Algebra",
"Definition:Lebesgue Measure",
"Definition:Complete Measure Space"
] | [
"Definition:Measure Space",
"Definition:Lebesgue Measure/Real Number Line",
"Measure Space from Outer Measure",
"Measure Space arising from Outer Measure is Complete",
"Definition:Complete Measure Space",
"Category:Lebesgue Measure"
] |
proofwiki-23073 | Norm of Bochner Measurable Function is Lebesgue Measurable | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\GF$.
Let $I$ be a real interval.
Let $f : I \to X$ be a Bochner measurable function.
Then $\norm f : I \to \R$ is Lebesgue measurable. | Since $f$ is a Bochner measurable function, there exists a sequence of simple functions $\sequence {s_n}_{n \mathop \in \N}$ with:
:$\ds \map f t = \lim_{n \mathop \to \infty} \map {s_n} t$ for almost all $t \in I$.
From Modulus of Limit: Normed Vector Space, we have:
:$\ds \norm {\map f t} = \lim_{n \mathop \to \infty... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $f : I \to X$ be a [[Definition:Bochner Measurable Function|Bochner measurable function]].
Then $\norm f : I \to \R$ is [[Defi... | Since $f$ is a [[Definition:Bochner Measurable Function|Bochner measurable function]], there exists a [[Definition:Sequence|sequence]] of [[Definition:Simple Function/Banach Space|simple functions]] $\sequence {s_n}_{n \mathop \in \N}$ with:
:$\ds \map f t = \lim_{n \mathop \to \infty} \map {s_n} t$ for [[Definition:Al... | Norm of Bochner Measurable Function is Lebesgue Measurable | https://proofwiki.org/wiki/Norm_of_Bochner_Measurable_Function_is_Lebesgue_Measurable | https://proofwiki.org/wiki/Norm_of_Bochner_Measurable_Function_is_Lebesgue_Measurable | [
"Bochner Measurable Functions"
] | [
"Definition:Banach Space",
"Definition:Real Interval",
"Definition:Bochner Measurable Function",
"Definition:Lebesgue Measurable Function"
] | [
"Definition:Bochner Measurable Function",
"Definition:Sequence",
"Definition:Simple Function/Banach Space",
"Definition:Almost All",
"Modulus of Limit/Normed Vector Space",
"Definition:Almost All",
"Left Composition of Banach Space-Valued Simple Function with Function Fixing Origin is Simple Function",
... |
proofwiki-23074 | Pettis Measurability Theorem | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $I$ be a real interval.
Let $f : I \to X$ be a function.
Then $f$ is Bochner measurable {{iff}} it is both weakly measurable and almost separately valued. | Let $\lambda$ be the Lebesgue measure.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$. | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $f : I \to X$ be a [[Definition:Function|function]].
Then $f$ is [[Definition:Bochner Measurable Function|Bochner measurable]... | Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]].
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,}_X}$. | Pettis Measurability Theorem | https://proofwiki.org/wiki/Pettis_Measurability_Theorem | https://proofwiki.org/wiki/Pettis_Measurability_Theorem | [
"Bochner Measurable Functions"
] | [
"Definition:Banach Space",
"Definition:Real Interval",
"Definition:Function",
"Definition:Bochner Measurable Function",
"Definition:Weakly Measurable Function",
"Definition:Almost Separately Valued Function"
] | [
"Definition:Lebesgue Measure",
"Definition:Normed Dual Space"
] |
proofwiki-23075 | Banach Space-Valued Function Constant on each Set in Countable Partition of Domain is Bochner Measurable | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $I$ be a real interval.
Let $\sequence {\Omega_n'}_{n \mathop \in \N}$ be a sequence of Lebesgue measurable subsets of $I$ such that:
:$\ds \bigcup_{n \mathop \in \N} \Omega_n' = I$
Let $f : I \to X$ be a function t... | Let $\lambda$ be the Lebesgue measure on $I$.
Define $c_n \in X$ by $f \sqbrk {\Omega_n'} = \set {c_n}$.
From Countable Union of Measurable Sets as Disjoint Union of Measurable Sets, there exists a pairwise disjoint sequence $\sequence {\Omega_n}_{n \mathop \in \N}$ of Lebesgue measurable subsets of $I$ such that:
:$\... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $\sequence {\Omega_n'}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Lebesgue Measurable Set|Lebesg... | Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] on $I$.
Define $c_n \in X$ by $f \sqbrk {\Omega_n'} = \set {c_n}$.
From [[Countable Union of Measurable Sets as Disjoint Union of Measurable Sets]], there exists a [[Definition:Pairwise Disjoint|pairwise disjoint]] [[Definition:Sequence|sequence]]... | Banach Space-Valued Function Constant on each Set in Countable Partition of Domain is Bochner Measurable | https://proofwiki.org/wiki/Banach_Space-Valued_Function_Constant_on_each_Set_in_Countable_Partition_of_Domain_is_Bochner_Measurable | https://proofwiki.org/wiki/Banach_Space-Valued_Function_Constant_on_each_Set_in_Countable_Partition_of_Domain_is_Bochner_Measurable | [
"Simple Functions",
"Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Real Interval",
"Definition:Sequence",
"Definition:Measurable Set/Subsets of Real Space",
"Definition:Function",
"Definition:Constant Mapping",
"Definition:Bochner Measurable Function"
] | [
"Definition:Lebesgue Measure",
"Countable Union of Measurable Sets as Disjoint Union of Measurable Sets",
"Definition:Pairwise Disjoint",
"Definition:Sequence",
"Definition:Measurable Set/Subsets of Real Space",
"Definition:Empty Set",
"Definition:Finite Set",
"Banach Space-Valued Simple Function is B... |
proofwiki-23076 | Coslice Category is Isomorphic to Identity Functor Under Object Comma Category/Lemma 1 | :$I = J_2 \circ I_1$ | For each object $g$ in $\mathbf C / C$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J_2 \circ I_1} g
| r = \map {J_2} {I_1 g}
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J_2 \tuple{\operatorname{dom} g, *, g}
| c = Definition of $I_1$
}}
{{eqn | r = \tuple{\operatorname{dom} g, g}
| c = Defin... | :$I = J_2 \circ I_1$ | For each [[Definition:Object (Category Theory)|object]] $g$ in $\mathbf C / C$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J_2 \circ I_1} g
| r = \map {J_2} {I_1 g}
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J_2 \tuple{\operatorname{dom} g, *, g}
| c = Definition of $I_1$
}}
{{eqn | r = \tuple{\o... | Coslice Category is Isomorphic to Identity Functor Under Object Comma Category/Lemma 1 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Identity_Functor_Under_Object_Comma_Category/Lemma_1 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Identity_Functor_Under_Object_Comma_Category/Lemma_1 | [
"Coslice Category is Isomorphic to Identity Functor Under Object Comma Category"
] | [] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Coslice Category is Isomorphic to Identity Functor Under Object Comma Category"
] |
proofwiki-23077 | Coslice Category is Isomorphic to Identity Functor Under Object Comma Category/Lemma 2 | :$J = J_1 \circ I_2$ | For each object $\tuple{D, f}$ in $\paren{C \downarrow \operatorname{id}_{\mathbf C} }$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J_1 \circ I_2} \tuple{D, f}
| r = \map {J_1} {I_2 \tuple{D, f} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J_1 \tuple{*, D, f}
| c = Definition of $I_2$
}}
{{eqn | r... | :$J = J_1 \circ I_2$ | For each [[Definition:Object (Category Theory)|object]] $\tuple{D, f}$ in $\paren{C \downarrow \operatorname{id}_{\mathbf C} }$ we have:
{{begin-eqn}}
{{eqn | l = \paren{J_1 \circ I_2} \tuple{D, f}
| r = \map {J_1} {I_2 \tuple{D, f} }
| c = {{Defof|Composite Functor}}
}}
{{eqn | r = J_1 \tuple{*, D, f}
... | Coslice Category is Isomorphic to Identity Functor Under Object Comma Category/Lemma 2 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Identity_Functor_Under_Object_Comma_Category/Lemma_2 | https://proofwiki.org/wiki/Coslice_Category_is_Isomorphic_to_Identity_Functor_Under_Object_Comma_Category/Lemma_2 | [
"Coslice Category is Isomorphic to Identity Functor Under Object Comma Category"
] | [] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Coslice Category is Isomorphic to Identity Functor Under Object Comma Category"
] |
proofwiki-23078 | Banach Space-Valued Simple Function is Bochner Measurable | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $I$ be a real interval.
Let $f : I \to X$ be a simple function.
Then $f$ is Bochner measurable. | Define $s_n = f$ for each $n \in \N$.
Then we have:
:$\ds \map f t = \lim_{n \mathop \to \infty} \map {s_n} t$ for each $t \in I$.
Since $s_n$ is simple for each $n \in \N$, we have that $f$ is Bochner measurable.
{{qed}}
Category:Bochner Measurable Functions
cc0e5k9jop4to24z35vhhjw68g1qjbp | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $f : I \to X$ be a [[Definition:Simple Function/Banach Space|simple function]].
Then $f$ is [[Definition:Bochner Measurable F... | Define $s_n = f$ for each $n \in \N$.
Then we have:
:$\ds \map f t = \lim_{n \mathop \to \infty} \map {s_n} t$ for each $t \in I$.
Since $s_n$ is [[Definition:Simple Function/Banach Space|simple]] for each $n \in \N$, we have that $f$ is [[Definition:Bochner Measurable Function|Bochner measurable]].
{{qed}}
[[Catego... | Banach Space-Valued Simple Function is Bochner Measurable | https://proofwiki.org/wiki/Banach_Space-Valued_Simple_Function_is_Bochner_Measurable | https://proofwiki.org/wiki/Banach_Space-Valued_Simple_Function_is_Bochner_Measurable | [
"Bochner Measurable Functions"
] | [
"Definition:Banach Space",
"Definition:Real Interval",
"Definition:Simple Function/Banach Space",
"Definition:Bochner Measurable Function"
] | [
"Definition:Simple Function/Banach Space",
"Definition:Bochner Measurable Function",
"Category:Bochner Measurable Functions"
] |
proofwiki-23079 | Simple Function is Bochner Integrable | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $I$ be a real interval.
Let $f : I \to X$ be a simple function.
Then $f$ is Bochner integrable. | Let $\lambda$ be the Lebesgue measure on $I$.
Define $s_n = f$ for each $n \in \N$.
Then $s_n$ is a simple function for each $n \in \N$.
We then have:
:$\ds \int_I \norm {\map f t - \map {s_n} t}_X \rd \map \lambda t = \int_I 0 \rd \map \lambda t = 0$
Hence:
:$\ds \lim_{n \mathop \to \infty} \int_I \norm {\map f t - \m... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $f : I \to X$ be a [[Definition:Simple Function/Banach Space|simple function]].
Then $f$ is [[Definition:Bochner Integrable F... | Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] on $I$.
Define $s_n = f$ for each $n \in \N$.
Then $s_n$ is a [[Definition:Simple Function/Banach Space|simple function]] for each $n \in \N$.
We then have:
:$\ds \int_I \norm {\map f t - \map {s_n} t}_X \rd \map \lambda t = \int_I 0 \rd \map \lam... | Simple Function is Bochner Integrable | https://proofwiki.org/wiki/Simple_Function_is_Bochner_Integrable | https://proofwiki.org/wiki/Simple_Function_is_Bochner_Integrable | [
"Bochner Integrals"
] | [
"Definition:Banach Space",
"Definition:Real Interval",
"Definition:Simple Function/Banach Space",
"Definition:Bochner Integrable Function"
] | [
"Definition:Lebesgue Measure",
"Definition:Simple Function/Banach Space",
"Definition:Bochner Integrable Function",
"Category:Bochner Integrals"
] |
proofwiki-23080 | Mass of Earth | The mass $M$ of Earth is given as:
:$M \approx 5 \cdotp 976 \times 10^{24} \, \mathrm {kg}$ | From Kepler's Third Law of Planetary Motion:
:$T^2 = \dfrac {4 \pi^2} {G M} a^3$
where:
:$T$ is the period of the orbit
:$G$ is the universal gravitational constant
:$M$ is the mass of the body being orbited
:$a$ is the length of the semimajor axis of the orbital ellipse
Though this is a law named for planetary motion,... | The [[Definition:Mass|mass]] $M$ of [[Definition:Earth|Earth]] is given as:
:$M \approx 5 \cdotp 976 \times 10^{24} \, \mathrm {kg}$ | From [[Kepler's Third Law of Planetary Motion]]:
:$T^2 = \dfrac {4 \pi^2} {G M} a^3$
where:
:$T$ is the [[Definition:Period of Orbit|period]] of the [[Definition:Orbit (Physics)|orbit]]
:$G$ is the [[Definition:Universal Gravitational Constant|universal gravitational constant]]
:$M$ is the [[Definition:Mass|mass]] of t... | Mass of Earth/Proof from Moon | https://proofwiki.org/wiki/Mass_of_Earth | https://proofwiki.org/wiki/Mass_of_Earth/Proof_from_Moon | [
"Mass of Earth",
"Earth",
"Mass"
] | [
"Definition:Mass",
"Definition:Earth"
] | [
"Kepler's Laws of Planetary Motion/Third Law",
"Definition:Orbit (Physics)/Period",
"Definition:Orbit (Physics)",
"Definition:Universal Gravitational Constant",
"Definition:Mass",
"Definition:Body",
"Definition:Orbit (Physics)",
"Definition:Length",
"Definition:Ellipse/Major Axis/Semimajor Axis",
... |
proofwiki-23081 | Bochner Integral of Scalar Multiple of Simple Function | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $I$ be a real interval.
Let $\lambda$ be the Lebesgue measure on $I$.
Let $\alpha \in \GF$.
Let $f : I \to X$ be a simple function.
Then:
:$\ds \int \paren {\alpha f} \rd \lambda = \alpha \int f \rd \lambda$
where ... | From Banach Space-Valued Simple Function has Standard Representation, there exists:
:disjoint Lebesgue measurable sets $E_1, \ldots, E_n \subseteq I$ of finite Lebesgue measure
:$x_1, \ldots, x_n \in X$
such that:
:$\ds \map f t = \sum_{r \mathop = 1}^n x_r \map {\chi_{E_r} } t$ for each $t \in I$.
We then have:
:$\ds ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] on $I$.
Let $\alpha \in \GF$.
Let $f : I \to X$ be a [[Def... | From [[Banach Space-Valued Simple Function has Standard Representation]], there exists:
:[[Definition:Disjoint Sets|disjoint]] [[Definition:Measurable Set|Lebesgue measurable sets]] $E_1, \ldots, E_n \subseteq I$ of [[Definition:Finite Extended Real Number|finite]] [[Definition:Lebesgue Measure|Lebesgue measure]]
:$x_1... | Bochner Integral of Scalar Multiple of Simple Function | https://proofwiki.org/wiki/Bochner_Integral_of_Scalar_Multiple_of_Simple_Function | https://proofwiki.org/wiki/Bochner_Integral_of_Scalar_Multiple_of_Simple_Function | [
"Bochner Integrals"
] | [
"Definition:Banach Space",
"Definition:Real Interval",
"Definition:Lebesgue Measure",
"Definition:Simple Function/Banach Space",
"Definition:Bochner Integral of Simple Function"
] | [
"Banach Space-Valued Simple Function has Standard Representation",
"Definition:Disjoint Sets",
"Definition:Measurable Set",
"Definition:Finite Extended Real Number",
"Definition:Lebesgue Measure",
"Category:Bochner Integrals"
] |
proofwiki-23082 | Norm of Bochner Integral of Simple Function | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $I$ be a real interval.
Let $\lambda$ be the Lebesgue measure on $I$.
Let $f : I \to X$ be a simple function.
Then:
:$\ds \norm {\int f \rd \lambda}_X \le \int \norm f_X \rd \lambda$
where the $\int$ on the {{LHS}}... | From Banach Space-Valued Simple Function has Standard Representation, there exists:
:disjoint Lebesgue measurable sets $E_1, \ldots, E_n \subseteq I$ of finite Lebesgue measure
:$x_1, \ldots, x_n \in X$
such that:
:$\ds \map f t = \sum_{r \mathop = 1}^n x_r \map {\chi_{E_r} } t$ for each $t \in I$.
Then:
:$\ds \int f \... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] on $I$.
Let $f : I \to X$ be a [[Definition:Simple Function... | From [[Banach Space-Valued Simple Function has Standard Representation]], there exists:
:[[Definition:Disjoint Sets|disjoint]] [[Definition:Measurable Set|Lebesgue measurable sets]] $E_1, \ldots, E_n \subseteq I$ of [[Definition:Finite Extended Real Number|finite]] [[Definition:Lebesgue Measure|Lebesgue measure]]
:$x_1... | Norm of Bochner Integral of Simple Function | https://proofwiki.org/wiki/Norm_of_Bochner_Integral_of_Simple_Function | https://proofwiki.org/wiki/Norm_of_Bochner_Integral_of_Simple_Function | [
"Bochner Integrals"
] | [
"Definition:Banach Space",
"Definition:Real Interval",
"Definition:Lebesgue Measure",
"Definition:Simple Function/Banach Space",
"Definition:Bochner Integral of Simple Function",
"Definition:Lebesgue Integral"
] | [
"Banach Space-Valued Simple Function has Standard Representation",
"Definition:Disjoint Sets",
"Definition:Measurable Set",
"Definition:Finite Extended Real Number",
"Definition:Lebesgue Measure",
"Left Composition of Banach Space-Valued Simple Function with Function Fixing Origin is Simple Function",
"... |
proofwiki-23083 | Bochner Integral is Well-Defined | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $I$ be a real interval.
Let $\lambda$ be the Lebesgue measure on $I$.
Let $f : I \to X$ be a Bochner integrable function.
Then the Bochner integral of $f$ is well-defined. | We have two aims.
We first show that if $\sequence {s_n}_{n \mathop \in \N}$ is a sequence of simple functions such that:
:$\map {s_n} t \to \map f t$ for each $t \in I \setminus \Omega_0$, $\map \lambda {\Omega_0} = 0$
and:
:$\ds \lim_{n \mathop \to \infty} \int_I \norm {f - s_n}_X \rd \lambda = 0$
then:
:$\ds \lim_{... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] on $I$.
Let $f : I \to X$ be a [[Definition:Bochner Integra... | We have two aims.
We first show that if $\sequence {s_n}_{n \mathop \in \N}$ is a [[Definition:Sequence|sequence]] of [[Definition:Simple Function/Banach Space|simple functions]] such that:
:$\map {s_n} t \to \map f t$ for each $t \in I \setminus \Omega_0$, $\map \lambda {\Omega_0} = 0$
and:
:$\ds \lim_{n \mathop \to... | Bochner Integral is Well-Defined | https://proofwiki.org/wiki/Bochner_Integral_is_Well-Defined | https://proofwiki.org/wiki/Bochner_Integral_is_Well-Defined | [
"Bochner Integrals"
] | [
"Definition:Banach Space",
"Definition:Real Interval",
"Definition:Lebesgue Measure",
"Definition:Bochner Integrable Function",
"Definition:Bochner Integral"
] | [
"Definition:Sequence",
"Definition:Simple Function/Banach Space",
"Definition:Limit of Sequence",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Simple Function/Banach Space",
"Bochner Integral of Sum of Simple Functions",
"Bochner Integral of Scalar Multiple of Simple Function",
"Norm of... |
proofwiki-23084 | Bochner Integral is Additive | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $I$ be a real interval.
Let $\lambda$ be the Lebesgue measure on $I$.
Let $f, g : I \to X$ be Bochner integrable functions.
Then $f + g$ is Bochner integrable and:
:$\ds \int \paren {f + g} \rd \lambda = \int f \rd... | Let $\sequence {u_n}_{n \mathop \in \N}$ be a sequence of simple functions such that:
:$\map {u_n} t \to \map f t$ for each $t \in I \setminus N_1$, $\map \lambda {N_1} = 0$
and:
:$\ds \lim_{n \mathop \to \infty} \int_I \norm {f - u_n}_X \rd \lambda = 0$
Then:
:$\ds \int_I f \rd \lambda = \lim_{n \mathop \to \infty} \i... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] on $I$.
Let $f, g : I \to X$ be [[Definition:Bochner Integr... | Let $\sequence {u_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Simple Function/Banach Space|simple functions]] such that:
:$\map {u_n} t \to \map f t$ for each $t \in I \setminus N_1$, $\map \lambda {N_1} = 0$
and:
:$\ds \lim_{n \mathop \to \infty} \int_I \norm {f - u_n}_X \rd \lambda = ... | Bochner Integral is Additive | https://proofwiki.org/wiki/Bochner_Integral_is_Additive | https://proofwiki.org/wiki/Bochner_Integral_is_Additive | [
"Bochner Integrals"
] | [
"Definition:Banach Space",
"Definition:Real Interval",
"Definition:Lebesgue Measure",
"Definition:Bochner Integrable Function",
"Definition:Bochner Integrable Function"
] | [
"Definition:Sequence",
"Definition:Simple Function/Banach Space",
"Definition:Sequence",
"Definition:Simple Function/Banach Space",
"Sum of Banach Space-Valued Simple Functions is Simple Function",
"Definition:Simple Function/Banach Space",
"Null Sets Closed under Countable Union",
"Definition:Null Se... |
proofwiki-23085 | Bochner Integral is Homogeneous | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.
Let $I$ be a real interval.
Let $\lambda$ be the Lebesgue measure on $I$.
Let $f : I \to X$ be a Bochner integrable function.
Let $\alpha \in \GF$.
Then $\alpha f$ is Bochner integrable and:
:$\ds \int_I \paren {\alpha... | Let $\sequence {u_n}_{n \mathop \in \N}$ be a sequence of simple functions such that:
:$\map {u_n} t \to \map f t$ for each $t \in I \setminus \Omega_0$, $\map \lambda {\Omega_0} = 0$
and:
:$\ds \lim_{n \mathop \to \infty} \int_I \norm {f - u_n}_X \rd \lambda = 0$
Then:
:$\alpha \map {u_n} t \to \alpha \map f t$ for ea... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $I$ be a [[Definition:Real Interval|real interval]].
Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] on $I$.
Let $f : I \to X$ be a [[Definition:Bochner Integra... | Let $\sequence {u_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Simple Function/Banach Space|simple functions]] such that:
:$\map {u_n} t \to \map f t$ for each $t \in I \setminus \Omega_0$, $\map \lambda {\Omega_0} = 0$
and:
:$\ds \lim_{n \mathop \to \infty} \int_I \norm {f - u_n}_X \rd ... | Bochner Integral is Homogeneous | https://proofwiki.org/wiki/Bochner_Integral_is_Homogeneous | https://proofwiki.org/wiki/Bochner_Integral_is_Homogeneous | [
"Bochner Integrals"
] | [
"Definition:Banach Space",
"Definition:Real Interval",
"Definition:Lebesgue Measure",
"Definition:Bochner Integrable Function",
"Definition:Bochner Integrable Function"
] | [
"Definition:Sequence",
"Definition:Simple Function/Banach Space",
"Definition:Bochner Integrable Function",
"Integral of Integrable Function is Homogeneous",
"Category:Bochner Integrals"
] |
proofwiki-23086 | Identity Natural Transformation is Natural Transformation | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F : \mathbf C \to \mathbf D$ be a covariant functor.
Let $\operatorname{id}_F: F \to F$ denote the identity natural transformation
Then:
:$\operatorname{id}_F$ is a natural transformation. | {{Recall|Identity Natural Transformation|identity natural transformation}}
{{:Definition:Identity Natural Transformation}}
By definition of identity morphsim for all morphisms $f: X \to Y$ in $\mathbf C$:
:$\operatorname{id}_{FY} \circ Ff = Ff = Ff \circ \operatorname{id}_{FX}$
Hence for all morphisms $f: X \to Y$ in $... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F : \mathbf C \to \mathbf D$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\operatorname{id}_F: F \to F$ denote the [[Definition:Identity Natural Transformation|identity natural transformation]]
Then:
:$\operatorname{id}_F$ ... | {{Recall|Identity Natural Transformation|identity natural transformation}}
{{:Definition:Identity Natural Transformation}}
By definition of [[Definition:Identity Morphism|identity morphsim]] for all [[Definition:Morphism (Category Theory)|morphisms]] $f: X \to Y$ in $\mathbf C$:
:$\operatorname{id}_{FY} \circ Ff = Ff ... | Identity Natural Transformation is Natural Transformation | https://proofwiki.org/wiki/Identity_Natural_Transformation_is_Natural_Transformation | https://proofwiki.org/wiki/Identity_Natural_Transformation_is_Natural_Transformation | [
"Identity Natural Transformations",
"Natural Transformations"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Identity Natural Transformation",
"Definition:Natural Transformation/Covariant Functors"
] | [
"Definition:Identity Morphism",
"Definition:Morphism",
"Definition:Morphism",
"Definition:Natural Transformation/Covariant Functors",
"Category:Identity Natural Transformations",
"Category:Natural Transformations"
] |
proofwiki-23087 | Identity Natural Transformation is Left Identity | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\eta: F \to G$ be a natural transformation from $F$ to $G$.
Let $\operatorname{id}_G$ denote the identity natural transformation on $G$.
Then:
:$\operatorname{id}_G \mathop \circ \eta = \eta$
where $\operator... | By definition of natural transformation for each object $X \in \mathbf C$:
:$\eta_X$ is a morphism from $F X$ to $G X$
such that for all objects $X, Y$ in $\mathbf C$ and for each morphism $f: X \to Y$ of $\mathbf C$:
:$\eta_Y \circ F f = G f \circ \eta_X$
For each object $X \in \mathbf C$ we have:
{{begin-eqn}}
{{eqn ... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta: F \to G$ be a [[Definition:Natural Transformation|natural transformation]] from $F$ to $G$.
Let $\operatorname{id}_G$ denote the [[Definition... | By definition of [[Definition:Natural Transformation|natural transformation]] for each [[Definition:Object (Category Theory)|object]] $X \in \mathbf C$:
:$\eta_X$ is a [[Definition:Morphism (Category Theory)|morphism]] from $F X$ to $G X$
such that for all [[Definition:Object (Category Theory)|objects]] $X, Y$ in $\mat... | Identity Natural Transformation is Left Identity | https://proofwiki.org/wiki/Identity_Natural_Transformation_is_Left_Identity | https://proofwiki.org/wiki/Identity_Natural_Transformation_is_Left_Identity | [
"Identity Natural Transformations"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Identity Natural Transformation",
"Definition:Vertical Composition of Natural Transformations"
] | [
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Theory)"
] |
proofwiki-23088 | Identity Natural Transformation is Right Identity | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\eta: F \to G$ be a natural transformation from $F$ to $G$.
Let $\operatorname{id}_F$ denote the identity natural transformation on $F$.
Then:
:$\eta \circ \operatorname{id}_F = \eta$ | By definition of natural transformation for each object $X \in \mathbf C$:
:$\eta_X$ is a morphism from $F X$ to $G X$
such that for all objects $X, Y$ in $\mathbf C$ and for each morphism $f: X \to Y$ of $\mathbf C$:
:$\eta_Y \circ F f = G f \circ \eta_X$
For each object $X \in \mathbf C$ we have:
{{begin-eqn}}
{{eqn ... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta: F \to G$ be a [[Definition:Natural Transformation|natural transformation]] from $F$ to $G$.
Let $\operatorname{id}_F$ denote the [[Definition... | By definition of [[Definition:Natural Transformation|natural transformation]] for each [[Definition:Object (Category Theory)|object]] $X \in \mathbf C$:
:$\eta_X$ is a [[Definition:Morphism (Category Theory)|morphism]] from $F X$ to $G X$
such that for all [[Definition:Object (Category Theory)|objects]] $X, Y$ in $\mat... | Identity Natural Transformation is Right Identity | https://proofwiki.org/wiki/Identity_Natural_Transformation_is_Right_Identity | https://proofwiki.org/wiki/Identity_Natural_Transformation_is_Right_Identity | [
"Identity Natural Transformations"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Identity Natural Transformation"
] | [
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Theory)"
] |
proofwiki-23089 | Identity Natural Transformation is Idempotent | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F : \mathbf C \to \mathbf D$ be a covariant functor.
Let $\operatorname{id}_F$ denote the identity natural transformation on $F$.
Then:
:$\operatorname{id}_F \circ \operatorname{id}_F = \operatorname{id}_F$
where $\operatorname{id}_F \circ \operatorname{id}_F$ denotes... | For each object $X$ in $\mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \paren{\operatorname{id}_F \circ \operatorname{id}_F}_X
| r = \paren{\operatorname{id}_F}_X \circ \paren{\operatorname{id}_F}_X
| c = {{Defof|Vertical Composition of Natural Transformations}}
}}
{{eqn | r = \operatorname{id}_{FX} \circ \o... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F : \mathbf C \to \mathbf D$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\operatorname{id}_F$ denote the [[Definition:Identity Natural Transformation|identity natural transformation]] on $F$.
Then:
:$\operatorname{id}_F \... | For each [[Definition:Object (Category Theory)|object]] $X$ in $\mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \paren{\operatorname{id}_F \circ \operatorname{id}_F}_X
| r = \paren{\operatorname{id}_F}_X \circ \paren{\operatorname{id}_F}_X
| c = {{Defof|Vertical Composition of Natural Transformations}}
}}
{{e... | Identity Natural Transformation is Idempotent | https://proofwiki.org/wiki/Identity_Natural_Transformation_is_Idempotent | https://proofwiki.org/wiki/Identity_Natural_Transformation_is_Idempotent | [
"Identity Natural Transformations"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Identity Natural Transformation",
"Definition:Vertical Composition of Natural Transformations"
] | [
"Definition:Object (Category Theory)",
"Identity Morphism is Idempotent",
"Category:Identity Natural Transformations"
] |
proofwiki-23090 | Identity Natural Transformation is Natural Isomorphism | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F : \mathbf C \to \mathbf D$ be a covariant functor.
Let $\operatorname{id}_F: F \to F$ denote the identity natural transformation
Then:
:$\operatorname{id}_F$ is a natural isomorphism | By definition of identity natural transformation:
:for each $X \in \mathbf C : \paren{\operatorname{id}_F}_X = \operatorname{id}_{F X}$
where $\operatorname{id}_{F X}$ is the identity morphism on $F X$.
From Identity Morphism is Isomorphism:
:for each $X \in \mathbf C : \operatorname{id}_{F X}$ is an isomorphism of $\m... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F : \mathbf C \to \mathbf D$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\operatorname{id}_F: F \to F$ denote the [[Definition:Identity Natural Transformation|identity natural transformation]]
Then:
:$\operatorname{id}_F$ ... | By definition of [[Definition:Identity Natural Transformation|identity natural transformation]]:
:for each $X \in \mathbf C : \paren{\operatorname{id}_F}_X = \operatorname{id}_{F X}$
where $\operatorname{id}_{F X}$ is the [[Definition:Identity Morphism|identity morphism]] on $F X$.
From [[Identity Morphism is Isomor... | Identity Natural Transformation is Natural Isomorphism | https://proofwiki.org/wiki/Identity_Natural_Transformation_is_Natural_Isomorphism | https://proofwiki.org/wiki/Identity_Natural_Transformation_is_Natural_Isomorphism | [
"Natural Isomorphisms",
"Identity Natural Transformations"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Identity Natural Transformation",
"Definition:Natural Isomorphism"
] | [
"Definition:Identity Natural Transformation",
"Definition:Identity Morphism",
"Identity Morphism is Isomorphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Definition:Natural Isomorphism",
"Category:Natural Isomorphisms",
"Category:Identity Natural Transf... |
proofwiki-23091 | Equivalence of Definitions of Adjunction | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$, $\mathbf D$ be locally small categories.
{{TFAE|def = Adjunction}} | By definition of natural isomorphism
:$\alpha: \map {\mathrm {Hom}_{\mathbf C} } {F-, -} \to \map {\mathrm {Hom}_{\mathbf D} } {-, G-}$ is a natural isomorphism between:
::the hom bifunctor with left functor $\map {\mathrm {Hom}_{\mathbf C} } {F-, -} : \mathbf D^{\mathrm {op}} \times \mathbf C \to \mathbf{Set}$
:and:
:... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
{{TFAE|def = Adjunction}} | By definition of [[Definition:Natural Isomorphism|natural isomorphism]]
:$\alpha: \map {\mathrm {Hom}_{\mathbf C} } {F-, -} \to \map {\mathrm {Hom}_{\mathbf D} } {-, G-}$ is a [[Definition:Natural Isomorphism|natural isomorphism]] between:
::the [[Definition:Hom Bifunctor With Left Functor|hom bifunctor with left func... | Equivalence of Definitions of Adjunction | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Adjunction | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Adjunction | [
"Adjunctions"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category"
] | [
"Definition:Natural Isomorphism",
"Definition:Natural Isomorphism",
"Definition:Hom Bifunctor With Left Functor",
"Definition:Hom Bifunctor With Right Functor",
"Definition:Object (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Morphis... |
proofwiki-23092 | Surjection iff Epimorphism in Category of Sets/Necessary Condition Proof 1 | Let $\mathbf{Set}$ be the category of sets.
Let $f: X \to Y$ be a surjection.
Then:
:$f$ is an epimorphism in $\mathbf{Set}$. | By definition of epimorphism:
:$f$ is an epimorphism
{{iff}}
:for all mappings $g, h: Y \to Z : g \circ f = h \circ f \implies g = h$
In the following we prove the contrapositive statement:
:for all mappings $g, h: Y \to Z : g \ne h \implies g \circ f \ne h \circ f$
Let $g, h: Y \to Z$ be mappings such that $g \ne h$.
... | Let $\mathbf{Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $f: X \to Y$ be a [[Definition:Surjection|surjection]].
Then:
:$f$ is an [[Definition:Epimorphism (Category Theory)|epimorphism]] in $\mathbf{Set}$. | By definition of [[Definition:Epimorphism (Category Theory)|epimorphism]]:
:$f$ is an [[Definition:Epimorphism (Category Theory)|epimorphism]]
{{iff}}
:for all [[Definition:Mapping|mappings]] $g, h: Y \to Z : g \circ f = h \circ f \implies g = h$
In the following we prove the [[Definition:Contrapositive Statement|con... | Surjection iff Epimorphism in Category of Sets/Necessary Condition Proof 1 | https://proofwiki.org/wiki/Surjection_iff_Epimorphism_in_Category_of_Sets/Necessary_Condition_Proof_1 | https://proofwiki.org/wiki/Surjection_iff_Epimorphism_in_Category_of_Sets/Necessary_Condition_Proof_1 | [
"Surjection iff Epimorphism in Category of Sets"
] | [
"Definition:Category of Sets",
"Definition:Surjection",
"Definition:Epimorphism (Category Theory)"
] | [
"Definition:Epimorphism (Category Theory)",
"Definition:Epimorphism (Category Theory)",
"Definition:Mapping",
"Definition:Contrapositive Statement",
"Definition:Mapping",
"Definition:Mapping",
"Equality of Mappings",
"Definition:Surjection",
"Equality of Mappings",
"Rule of Transposition",
"Defi... |
proofwiki-23093 | Surjection iff Epimorphism in Category of Sets/Necessary Condition Proof 2 | Let $\mathbf{Set}$ be the category of sets.
Let $f: X \to Y$ be a surjection.
Then:
:$f$ is an epimorphism in $\mathbf{Set}$. | Let $g, h: Y \to Z$ be mappings such that $g \circ f = h \circ f$.
Let $y \in Y$ be an arbitrary element of $Y$.
By definition of surjection:
:$(1) \quad \exists x \in X : \map f x = y$.
We have:
{{begin-eqn}}
{{eqn | l = \map g y
| r = \map g {\map f x}
| c = By $(1)$
}}
{{eqn | r = \map {\paren{g \circ f}... | Let $\mathbf{Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $f: X \to Y$ be a [[Definition:Surjection|surjection]].
Then:
:$f$ is an [[Definition:Epimorphism (Category Theory)|epimorphism]] in $\mathbf{Set}$. | Let $g, h: Y \to Z$ be [[Definition:Mapping|mappings]] such that $g \circ f = h \circ f$.
Let $y \in Y$ be an arbitrary [[Definition:Element|element]] of $Y$.
By definition of [[Definition:Surjection|surjection]]:
:$(1) \quad \exists x \in X : \map f x = y$.
We have:
{{begin-eqn}}
{{eqn | l = \map g y
| r = \... | Surjection iff Epimorphism in Category of Sets/Necessary Condition Proof 2 | https://proofwiki.org/wiki/Surjection_iff_Epimorphism_in_Category_of_Sets/Necessary_Condition_Proof_2 | https://proofwiki.org/wiki/Surjection_iff_Epimorphism_in_Category_of_Sets/Necessary_Condition_Proof_2 | [
"Surjection iff Epimorphism in Category of Sets"
] | [
"Definition:Category of Sets",
"Definition:Surjection",
"Definition:Epimorphism (Category Theory)"
] | [
"Definition:Mapping",
"Definition:Element",
"Definition:Surjection",
"Equality of Mappings",
"Definition:Epimorphism (Category Theory)"
] |
proofwiki-23094 | Surjection iff Epimorphism in Category of Sets/Sufficient Condition | Let $\mathbf{Set}$ be the category of sets.
Let $f: X \to Y$ be an epimorphism in $\mathbf{Set}$.
Then:
:$f$ is a surjection. | The contrapositive statement will be proved, that is:
:if $f$ is not a surjection then $f$ is not an epimorphism.
The result will follow by the Rule of Transposition.
Let $f$ be not a surjection.
By definition of surjection:
:$\exists y_0 \in Y : \forall x \in X: \map f x \ne y_0$
Consider the mappings defined by:
:$g:... | Let $\mathbf{Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $f: X \to Y$ be an [[Definition:Epimorphism (Category Theory)|epimorphism]] in $\mathbf{Set}$.
Then:
:$f$ is a [[Definition:Surjection|surjection]]. | The [[Definition:Contrapositive Statement|contrapositive statement]] will be proved, that is:
:if $f$ is not a [[Definition:Surjection|surjection]] then $f$ is not an [[Definition:Epimorphism (Category Theory)|epimorphism]].
The result will follow by the [[Rule of Transposition]].
Let $f$ be not a [[Definition:Surje... | Surjection iff Epimorphism in Category of Sets/Sufficient Condition | https://proofwiki.org/wiki/Surjection_iff_Epimorphism_in_Category_of_Sets/Sufficient_Condition | https://proofwiki.org/wiki/Surjection_iff_Epimorphism_in_Category_of_Sets/Sufficient_Condition | [
"Surjection iff Epimorphism in Category of Sets"
] | [
"Definition:Category of Sets",
"Definition:Epimorphism (Category Theory)",
"Definition:Surjection"
] | [
"Definition:Contrapositive Statement",
"Definition:Surjection",
"Definition:Epimorphism (Category Theory)",
"Rule of Transposition",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Mapping",
"Equality of Mappings",
"Equality of Mappings",
"Definition:Epimorphism (Category Theory)",
... |
proofwiki-23095 | Bijection Iff Isomorphism in Category of Sets | Let $\mathbf{Set}$ be the category of sets.
Let $f: X \to Y$ be a morphism in $\mathbf{Set}$, i.e. a mapping.
Then:
:$f$ is a bijection {{iff}} it is an isomorphism in $\mathbf{Set}$. | By definition of bijection:
:$f$ is a bijection
{{iff}}
:$f$ admits an inverse
By definition of inverse:
:$f$ admits an inverse
{{iff}}
:$\exists$ a mapping $g: Y \to X : g \circ f = I_X \land f \circ g = I_Y$
where:
:$I_X$ and $I_Y$ denote the identity mappings on $X$ and $Y$ respectively.
By definition of category... | Let $\mathbf{Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $f: X \to Y$ be a [[Definition:Morphism (Category Theory)|morphism]] in $\mathbf{Set}$, i.e. a [[Definition:Mapping|mapping]].
Then:
:$f$ is a [[Definition:Bijection|bijection]] {{iff}} it is an [[Definition:Isomorphism (Category Theory)... | By definition of [[Definition:Bijection|bijection]]:
:$f$ is a [[Definition:Bijection|bijection]]
{{iff}}
:$f$ admits an [[Definition:Inverse Mapping|inverse]]
By definition of [[Definition:Inverse Mapping|inverse]]:
:$f$ admits an [[Definition:Inverse Mapping|inverse]]
{{iff}}
:$\exists$ a [[Definition:Mapping|ma... | Bijection Iff Isomorphism in Category of Sets | https://proofwiki.org/wiki/Bijection_Iff_Isomorphism_in_Category_of_Sets | https://proofwiki.org/wiki/Bijection_Iff_Isomorphism_in_Category_of_Sets | [
"Bijections",
"Category of Sets"
] | [
"Definition:Category of Sets",
"Definition:Morphism",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Isomorphism (Category Theory)"
] | [
"Definition:Bijection",
"Definition:Bijection",
"Definition:Inverse Mapping",
"Definition:Inverse Mapping",
"Definition:Inverse Mapping",
"Definition:Mapping",
"Definition:Identity Mapping",
"Definition:Category of Sets",
"Definition:Mapping",
"Definition:Morphism",
"Definition:Identity Morphism... |
proofwiki-23096 | Eccentricity of Ellipse in terms of Semiminor Axis and Linear Eccentricity | Let $K$ be a ellipse such that:
:$b$ denotes the length of semi-minor axis of $K$
:$c$ denotes the linear eccentricity of $K$
:$e$ denotes the eccentricity of $K$.
Then:
:$e = \dfrac c {\sqrt {b^2 + c^2} }$ | {{begin-eqn}}
{{eqn | l = e
| r = \dfrac c a
| c = Eccentricity of Ellipse is Linear Eccentricity over Semimajor Axis
}}
{{eqn | r = \dfrac c {\sqrt {b^2 + c^2} }
| c = Linear Eccentricity of Ellipse from Major and Minor Axis: $a^2 = b^2 + c^2$
}}
{{end-eqn}}
{{qed}}
Category:Eccentricity of Ellipse
C... | Let $K$ be a [[Definition:Ellipse|ellipse]] such that:
:$b$ denotes the [[Definition:Length (Linear Measure)|length]] of [[Definition:Semiminor Axis of Ellipse|semi-minor axis]] of $K$
:$c$ denotes the [[Definition:Linear Eccentricity|linear eccentricity]] of $K$
:$e$ denotes the [[Definition:Eccentricity of Ellipse|ec... | {{begin-eqn}}
{{eqn | l = e
| r = \dfrac c a
| c = [[Eccentricity of Ellipse is Linear Eccentricity over Semimajor Axis]]
}}
{{eqn | r = \dfrac c {\sqrt {b^2 + c^2} }
| c = [[Linear Eccentricity of Ellipse from Major and Minor Axis]]: $a^2 = b^2 + c^2$
}}
{{end-eqn}}
{{qed}}
[[Category:Eccentricity o... | Eccentricity of Ellipse in terms of Semiminor Axis and Linear Eccentricity | https://proofwiki.org/wiki/Eccentricity_of_Ellipse_in_terms_of_Semiminor_Axis_and_Linear_Eccentricity | https://proofwiki.org/wiki/Eccentricity_of_Ellipse_in_terms_of_Semiminor_Axis_and_Linear_Eccentricity | [
"Eccentricity of Ellipse",
"Linear Eccentricity",
"Semiminor Axis of Ellipse",
"Ellipses"
] | [
"Definition:Ellipse",
"Definition:Linear Measure/Length",
"Definition:Ellipse/Minor Axis/Semiminor Axis",
"Definition:Linear Eccentricity",
"Definition:Ellipse/Eccentricity"
] | [
"Eccentricity of Ellipse is Linear Eccentricity over Semimajor Axis",
"Linear Eccentricity of Ellipse from Major and Minor Axis",
"Category:Eccentricity of Ellipse",
"Category:Linear Eccentricity",
"Category:Semiminor Axis of Ellipse",
"Category:Ellipses"
] |
proofwiki-23097 | Coordinates of Foci of Ellipse in Reduced Form | Let $\KK$ be an ellipse embedded in a Cartesian plane in reduced form.
Then the foci of $\KK$ are at $\tuple {a e, 0}$ and $\tuple {-a e, 0}$, where:
:$a$ denotes the semimajor axis of $\KK$
:$e$ denotes the eccentricity of $\KK$. | By definition of the reduced form of $\KK$:
:the major axis of $\KK$ coincides with the $x$-axis
:the center of $\KK$ coincides with the origin.
Hence the foci of $\KK$ are themselves situated on the $x$-axis, equidistant from the origin $\tuple {0, 0}$.
Let $c$ denote the linear eccentricity of $\KK$.
Thus, by definit... | Let $\KK$ be an [[Definition:Ellipse|ellipse]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]] in [[Definition:Reduced Form of Ellipse|reduced form]].
Then the [[Definition:Focus of Ellipse|foci]] of $\KK$ are at $\tuple {a e, 0}$ and $\tuple {-a e, 0}$, where:
:$a$ denotes the [[Definition:Semimajor Axis... | By definition of the [[Definition:Reduced Form of Ellipse|reduced form]] of $\KK$:
:the [[Definition:Major Axis of Ellipse|major axis]] of $\KK$ coincides with the [[Definition:X-Axis|$x$-axis]]
:the [[Definition:Center of Ellipse|center]] of $\KK$ coincides with the [[Definition:Origin|origin]].
Hence the [[Definitio... | Coordinates of Foci of Ellipse in Reduced Form | https://proofwiki.org/wiki/Coordinates_of_Foci_of_Ellipse_in_Reduced_Form | https://proofwiki.org/wiki/Coordinates_of_Foci_of_Ellipse_in_Reduced_Form | [
"Reduced Form of Ellipse",
"Foci of Ellipses"
] | [
"Definition:Ellipse",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Ellipse",
"Definition:Ellipse/Focus",
"Definition:Ellipse/Major Axis/Semimajor Axis",
"Definition:Ellipse/Eccentricity"
] | [
"Definition:Conic Section/Reduced Form/Ellipse",
"Definition:Ellipse/Major Axis",
"Definition:Axis/X-Axis",
"Definition:Ellipse/Center",
"Definition:Coordinate System/Origin",
"Definition:Ellipse/Focus",
"Definition:Axis/X-Axis",
"Definition:Coordinate System/Origin",
"Definition:Linear Eccentricity... |
proofwiki-23098 | Equations of Directrices of Ellipse in Reduced Form | Let $\KK$ be an ellipse embedded in a Cartesian plane in reduced form.
Then the directrices of $\KK$ are the lines:
:$x = \dfrac a e$
and:
:$x = -\dfrac a e$
where:
:$a$ denotes the semimajor axis of $\KK$
:$e$ denotes the eccentricity of $\KK$. | By definition of the reduced form of $\KK$:
:the major axis of $\KK$ coincides with the $x$-axis
:the center of $\KK$ coincides with the origin.
By Directrix of Ellipse is Perpendicular to Major Axis, the directrices of $\KK$ are perpendicular to the $x$-axis.
Hence the equations of the directrices are in the form $x =... | Let $\KK$ be an [[Definition:Ellipse|ellipse]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]] in [[Definition:Reduced Form of Ellipse|reduced form]].
Then the [[Definition:Directrix of Ellipse|directrices]] of $\KK$ are the [[Definition:Straight Line|lines]]:
:$x = \dfrac a e$
and:
:$x = -\dfrac a e$
whe... | By definition of the [[Definition:Reduced Form of Ellipse|reduced form]] of $\KK$:
:the [[Definition:Major Axis of Ellipse|major axis]] of $\KK$ coincides with the [[Definition:X-Axis|$x$-axis]]
:the [[Definition:Center of Ellipse|center]] of $\KK$ coincides with the [[Definition:Origin|origin]].
By [[Directrix of Ell... | Equations of Directrices of Ellipse in Reduced Form | https://proofwiki.org/wiki/Equations_of_Directrices_of_Ellipse_in_Reduced_Form | https://proofwiki.org/wiki/Equations_of_Directrices_of_Ellipse_in_Reduced_Form | [
"Reduced Form of Ellipse",
"Directrices of Ellipses"
] | [
"Definition:Ellipse",
"Definition:Cartesian Plane",
"Definition:Conic Section/Reduced Form/Ellipse",
"Definition:Ellipse/Directrix",
"Definition:Line/Straight Line",
"Definition:Ellipse/Major Axis/Semimajor Axis",
"Definition:Ellipse/Eccentricity"
] | [
"Definition:Conic Section/Reduced Form/Ellipse",
"Definition:Ellipse/Major Axis",
"Definition:Axis/X-Axis",
"Definition:Ellipse/Center",
"Definition:Coordinate System/Origin",
"Directrix of Ellipse is Perpendicular to Major Axis",
"Definition:Ellipse/Directrix",
"Definition:Right Angle/Perpendicular",... |
proofwiki-23099 | Identity Morphism is Idempotent | Let $\mathbf C$ be a category.
Let $X$ be an object of $\mathbf C$.
Let $\operatorname{id}_X : X \to X$ denote the identity morphism of $X$.
Then:
:$\operatorname{id}_X \circ \operatorname{id}_X = \operatorname{id}_X$ | By definition of identity morphism:
:for each object $Y \in \mathbf C$ and morphism $f : X \to Y$ in $\mathbf C$:
::$f \circ \operatorname{id}_X = f$
In particular, substituting $\operatorname{id}_X$ for $f$:
:$\operatorname{id}_X \circ \operatorname{id}_X = \operatorname{id}_X$
{{qed}}
Category:Morphisms
0h7c1ovv12jxs... | Let $\mathbf C$ be a [[Definition:Category|category]].
Let $X$ be an [[Definition:Object|object]] of $\mathbf C$.
Let $\operatorname{id}_X : X \to X$ denote the [[Definition:Identity Morphism|identity morphism]] of $X$.
Then:
:$\operatorname{id}_X \circ \operatorname{id}_X = \operatorname{id}_X$ | By definition of [[Definition:Identity Morphism|identity morphism]]:
:for each [[Definition:Object (Category Theory)|object]] $Y \in \mathbf C$ and [[Definition:Morphism (Category Theory)|morphism]] $f : X \to Y$ in $\mathbf C$:
::$f \circ \operatorname{id}_X = f$
In particular, substituting $\operatorname{id}_X$ for... | Identity Morphism is Idempotent | https://proofwiki.org/wiki/Identity_Morphism_is_Idempotent | https://proofwiki.org/wiki/Identity_Morphism_is_Idempotent | [
"Morphisms"
] | [
"Definition:Category",
"Definition:Object",
"Definition:Identity Morphism"
] | [
"Definition:Identity Morphism",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Category:Morphisms"
] |
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