From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Wed, 14 Feb 2024 07:24:38 +0100 Subject: Update --- .../fixed-point-theorems/index.md | 13 +++ pages/more-functional-analysis/index.md | 8 ++ .../alaoglu-bourbaki-theorem.md | 28 ++++++ .../locally-convex-spaces/index.md | 9 ++ .../locally-convex-spaces/krein-milman-theorem.md | 19 ++++ .../topological-vector-spaces/index.md | 56 +++++++++++ .../topological-vector-spaces/polar-topologies.md | 109 +++++++++++++++++++++ 7 files changed, 242 insertions(+) create mode 100644 pages/more-functional-analysis/fixed-point-theorems/index.md create mode 100644 pages/more-functional-analysis/index.md create mode 100644 pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md create mode 100644 pages/more-functional-analysis/locally-convex-spaces/index.md create mode 100644 pages/more-functional-analysis/locally-convex-spaces/krein-milman-theorem.md create mode 100644 pages/more-functional-analysis/topological-vector-spaces/index.md create mode 100644 pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md (limited to 'pages/more-functional-analysis') diff --git a/pages/more-functional-analysis/fixed-point-theorems/index.md b/pages/more-functional-analysis/fixed-point-theorems/index.md new file mode 100644 index 0000000..b90bf00 --- /dev/null +++ b/pages/more-functional-analysis/fixed-point-theorems/index.md @@ -0,0 +1,13 @@ +--- +title: Fixed-Point Theorems +parent: More Functional Analysis +nav_order: 1 +has_children: true +has_toc: false +--- + +# {{ page.title }} + +{% theorem * Banach Fixed-Point Theorem %} +test +{% endtheorem %} diff --git a/pages/more-functional-analysis/index.md b/pages/more-functional-analysis/index.md new file mode 100644 index 0000000..1af344d --- /dev/null +++ b/pages/more-functional-analysis/index.md @@ -0,0 +1,8 @@ +--- +title: More Functional Analysis +nav_order: 4 +has_children: true +has_toc: false +--- + +# {{ page.title }} diff --git a/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md b/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md new file mode 100644 index 0000000..be424c3 --- /dev/null +++ b/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md @@ -0,0 +1,28 @@ +--- +title: Alaoglu–Bourbaki Theorem +parent: Locally Convex Spaces +grand_parent: More Functional Analysis +nav_order: 1 +--- + +# {{ page.title }} + +Let $X$ be locally convex space and +let $U \subset X$ be a neighborhood of zero. +Let $X'$ denote the continuous dual of $X$. +Recall that there is a canonical pairing + +$$ +X \times X' \to \CC, \quad (x,f) \mapsto \angles{x,f} = f(x). +$$ + +The weak topology on $X'$ with respect to this pairing +is called weak\* topology. +It is the weakest topology on $X'$ such that +all evaluation maps $\angles{x,\cdot} : X \to \CC$ are continuous. +The polar of $U$ is the subset $U^{\circ} \subset X'$. +The theorem asserts that $U^{\circ}$ is compact in the weak\* topology. + +{% theorem * Alaoglu–Bourbaki Theorem %} +The polar of a neighborhood of zero in a locally convex space is weak\* compact. +{% endtheorem %} diff --git a/pages/more-functional-analysis/locally-convex-spaces/index.md b/pages/more-functional-analysis/locally-convex-spaces/index.md new file mode 100644 index 0000000..990ec99 --- /dev/null +++ b/pages/more-functional-analysis/locally-convex-spaces/index.md @@ -0,0 +1,9 @@ +--- +title: Locally Convex Spaces +parent: More Functional Analysis +nav_order: 2 +has_children: true +has_toc: false +--- + +# {{ page.title }} diff --git a/pages/more-functional-analysis/locally-convex-spaces/krein-milman-theorem.md b/pages/more-functional-analysis/locally-convex-spaces/krein-milman-theorem.md new file mode 100644 index 0000000..58b9182 --- /dev/null +++ b/pages/more-functional-analysis/locally-convex-spaces/krein-milman-theorem.md @@ -0,0 +1,19 @@ +--- +title: Krein–Milman Theorem +parent: Locally Convex Spaces +grand_parent: More Functional Analysis +nav_order: 2 +--- + +# {{ page.title }} + +## Extreme Points + +{% definition Extreme Point %} +Suppose $C$ is a convex subset of a vector space $X$. +We say that an element $x \in C$ is an *extreme point* of $C$ +if +{% enddefinition %} + +{% proof %} +{% endproof %} diff --git a/pages/more-functional-analysis/topological-vector-spaces/index.md b/pages/more-functional-analysis/topological-vector-spaces/index.md new file mode 100644 index 0000000..745d53b --- /dev/null +++ b/pages/more-functional-analysis/topological-vector-spaces/index.md @@ -0,0 +1,56 @@ +--- +title: Topological Vector Spaces +parent: More Functional Analysis +nav_order: 1 +has_children: true +has_toc: false +--- + +# {{ page.title }} + +Let $X$ be a set. +A *property* of subsets of $X$ is a set $P \subset \mathcal{P}(X)$. +We say that a subset $A \subset X$ has the property $P$, if $A \in P$. +A property $P$ of subsets of $X$ is said to be *stable under arbitrary intersections*, +if for every family $F$ of subsets of $X$ with property $P$, +the intersection $\bigcap F$ has the property $P$. +In other words, $P$ is stable under arbitrary intersections iff +$\bigcap F \in P$ for every subset $F \subset P$. +In this definition the family $F$ is allowed to be empty, +hence $\bigcap \emptyset = X$ needs to have the property $P$. + +For example, in a topological space $X$ the property of being a closed subset of $X$ +is stable under arbitrary intersections. + +If $P$ is stable under arbitrary intersections, +and $A$ is a subset of $X$, which may or may not have the property $P$, +then we define the *$P$-hull* of $A$ to be +the intersection of all supersets $B \supset A$ +having have the property $P$. +By definition, the $P$-hull of $A$ has the property $P$. +Moreover, it is the smallest superset of $A$ with property $P$ +in the following sense: If $B$ is any superset of $A$ with property $P$, +then $B$ contains the $P$-hull of $A$. + +For example, the "closed"-hull of a subset $A$ of a topological space +is the closure of $A$. + +There are the dual notions of being *stable under arbitrary unions* +and *$P$-core* with obvious definitions. + +{% definition Convex, Balanced, Absolutely Convex %} +Let $X$ be a vector space over the field $\KK$. +A subset $A \subset X$ is said to be +- *convex* if +- *balanced* if +- *absolutely convex* if +{% enddefinition %} + +{% theorem %} +These properties of subsets of $X$ +are stable under arbitrary intersections. +Thus we obtain the notions of +*convex hull* $\co A$, +*balanced hull* $\bal A$, and +*absolutely convex hull* $\aco A$. +{% endtheorem %} diff --git a/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md new file mode 100644 index 0000000..277ecd3 --- /dev/null +++ b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md @@ -0,0 +1,109 @@ +--- +title: Polar Topologies +parent: Topological Vector Spaces +grand_parent: More Functional Analysis +nav_order: 1 +--- + +# {{ page.title }} + +# Dual pairs of vector spaces + +Recall that a *bilinear form* on two vector spaces $V$ and $W$ over a field $\KK$ +is a mapping $b : V \times W \to \KK$ which is linear in each of its arguments, +that is, which satisfies + +$$ +\begin{align*} +b(v_1+v_2,w) &= b(v_1,w) + b(v_2,w) & +b(v,w_1+w_2) &= b(v,w_1) + b(v,w_2) \\ +b(\lambda v, w) &= \lambda \, b(v,w) & +b(v, \lambda w) &= \lambda \, b(v,w) +\end{align*} +$$ + +for all vectors $v,v_1,v_2 \in V$, $w,w_1,w_2 \in W$ and all scalars $\lambda \in \KK$. + +We say that the bilinear form $b : V \times W \to \KK$ is *nondegenerate*, if it has the properties + +$$ +\begin{gather*} +\forall v \in V : \qquad ( \forall w \in W : \angles{v,w} = 0 ) \implies v = 0 \\ +\forall w \in W : \qquad ( \forall v \in V : \angles{v,w} = 0 ) \implies w = 0 +\end{gather*} +$$ + +If $V$ is a vector space over $\KK$, +let us denote its *algebraic dual* by $V^*$. +Given a bilinear form $V \times W \to \KK$, consider the mappings + +$$ +c : V \to W*, c(v)(w) = b(v,w) +\tilde{c} : W \to V*, \tilde{c}(w)(v) = b(v,w) +$$ + +Then $b$ is nondegenerate if and only if +both $c$ and $\tilde{c}$ are injective. + + +{% definition Dual Pair %} +A *dual pair* (or *dual system* or *duality*) $\angles{V,W}$ over a field $\KK$ is constituted by +two vector spaces $V$ and $W$ over $\KK$ +and a nondegenerate bilinear form $\angles{\cdot,\cdot} : V \times W \to \KK$. +{% enddefinition %} + +(We resist saying that a dual pair is a triple ...) + +{% definition Weak Topology %} +Suppose $\angles{X,Y}$ is a dual pair of vector spaces over a field $\KK$. +We define the *weak topology on $X$*, denoted by $\sigma(X,Y)$, as +the [initial topology](/pages/general-topology/universal-constructions.html#initial-topology) induced by the maps +$\angles{\cdot,y} : X \to \KK$, where $y \in Y$. +Similarly, the *weak topology on $Y$*, denoted by $\sigma(Y,X)$, is +the initial topology induced by the maps +$\angles{x,\cdot} : Y \to \KK$, where $x \in X$. +{% enddefinition %} + +{% theorem Weak Topologies are Locally Convex %} +Suppose $\angles{X,Y}$ is a dual pair of vector spaces over a field $\KK$. +TODO +{% endtheorem %} + +## The Canonical Pairing + +TODO: Def & Theorem (weak rep) + +{% definition Polar Set %} +Suppose $\angles{X,Y}$ is a dual pair of vector spaces. +The *polar* of a subset $A \subset X$ is the set + +$$ +A^{\circ} = \braces{y \in Y : \abs{\angles{x,y}} \le 1 \ \forall x \in A}. +$$ + +The *polar* of a subset $B \subset Y$ is the set + +$$ +B^{\circ} = \braces{x \in X : \abs{\angles{x,y}} \le 1 \ \forall y \in B}. +$$ +{% enddefinition %} + +Some authors define the polar with the condition $\Re \angles{x,y} \le 1$ +instead of $\abs{\angles{x,y}} \le 1$ and call *absolute polar* what we call polar. +Some authors write $B_{\circ}$ for $B^{\circ}$. + +Note that the *bipolar* $A^{\circ\circ} = (A^{\circ})^{\circ}$ is a subset of $X$. + +{% theorem * Bipolar Theorem %} +Suppose $\angles{X,Y}$ is a dual pair of vector spaces +and $A \subset X$. Then + +$$ +A^{\circ\circ} = \overline{\aco(A)}, +$$ + +where the closure is taken with respect to the weak topology on $X$, that is $\sigma(X,Y)$. +{% endtheorem %} + +{% proof %} +{% endproof %} -- cgit v1.2.3-54-g00ecf