# Polar Topologies

## Dual pairs of vector spaces

Recall that a *bilinear form* on two vector spaces $V$ and $W$ over a field $K$ is a mapping $b:V×W→K$ which is linear in each of its arguments, that is, which satisfies

for all vectors $v,v_{1},v_{2}∈V$, $w,w_{1},w_{2}∈W$ and all scalars $λ∈K$.

We say that the bilinear form $b:V×W→K$ is *nondegenerate*, if it has the properties

If $V$ is a vector space over $K$, let us denote its *algebraic dual* by $V_{∗}$. Given a bilinear form $V×W→K$, consider the mappings

Then $b$ is nondegenerate if and only if both $c$ and $c~$ are injective.

Definition (Dual Pair)

A

dual pair(ordual systemorduality) $⟨V,W⟩$ over a field $K$ is constituted by two vector spaces $V$ and $W$ over $K$ and a nondegenerate bilinear form $⟨⋅,⋅⟩:V×W→K$.

(We resist saying that a dual pair is a triple …)

Definition (Weak Topology)

Suppose $⟨X,Y⟩$ is a dual pair of vector spaces over a field $K$. We define the

weak topology on $X$, denoted by $σ(X,Y)$, as the initial topology induced by the maps $⟨⋅,y⟩:X→K$, where $y∈Y$. Similarly, theweak topology on $Y$, denoted by $σ(Y,X)$, is the initial topology induced by the maps $⟨x,⋅⟩:Y→K$, where $x∈X$.

Theorem (Weak Topologies are Locally Convex)

Suppose $⟨X,Y⟩$ is a dual pair of vector spaces over a field $K$. TODO

## The Canonical Pairing

TODO: Def & Theorem (weak rep)

Definition (Polar Set)

Suppose $⟨X,Y⟩$ is a dual pair of vector spaces. The

$A_{∘}={y∈Y:∣⟨x,y⟩∣≤1∀x∈A}.$polarof a subset $A⊂X$ is the setThe

$B_{∘}={x∈X:∣⟨x,y⟩∣≤1∀y∈B}.$polarof a subset $B⊂Y$ is the set

Some authors define the polar with the condition $Re⟨x,y⟩≤1$ instead of $∣⟨x,y⟩∣≤1$ and call *absolute polar* what we call polar. Some authors write $B_{∘}$ for $B_{∘}$.

Note that the *bipolar* $A_{∘∘}=(A_{∘})_{∘}$ is a subset of $X$.

Bipolar Theorem

Suppose $⟨X,Y⟩$ is a dual pair of vector spaces and $A⊂X$. Then

$A_{∘∘}=aco(A) ,$where the closure is taken with respect to the weak topology on $X$, that is $σ(X,Y)$.