From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Wed, 14 Feb 2024 07:24:38 +0100
Subject: Update

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+---
+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 %}
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