---
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 %}