Polar Topologies

Dual pairs of vector spaces

Recall that a bilinear form on two vector spaces $V$ and $W$ over a field $\mathbb{K}$ is a mapping $b:V\times W\to \mathbb{K}$ which is linear in each of its arguments, that is, which satisfies

$$\begin{array}{rlrl}b(v_1+v_2,w)& =b(v_1,w)+b(v_2,w)& b(v,w_1+w_2)& =b(v,w_1)+b(v,w_2)\\ b(\lambda v,w)& =\lambda b(v,w)& b(v,\lambda w)& =\lambda b(v,w)\end{array}$$

for all vectors $v,v_1,v_2\in V$, $w,w_1,w_2\in W$ and all scalars $\lambda \in \mathbb{K}$.

We say that the bilinear form $b:V\times W\to \mathbb{K}$ is nondegenerate, if it has the properties

$$\begin{array}{c}\forall v\in V:(\forall w\in W:⟨v,w⟩=0)⟹v=0\\ \forall w\in W:(\forall v\in V:⟨v,w⟩=0)⟹w=0\end{array}$$

If $V$ is a vector space over $\mathbb{K}$, let us denote its algebraic dual by $V^{\ast }$. Given a bilinear form $V\times W\to \mathbb{K}$, consider the mappings

$$c:V\to W\ast ,c(v)(w)=b(v,w)\tilde{c}:W\to V\ast ,\tilde{c}(w)(v)=b(v,w)$$

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) $⟨V,W⟩$ over a field $\mathbb{K}$ is constituted by two vector spaces $V$ and $W$ over $\mathbb{K}$ and a nondegenerate bilinear form $⟨\cdot ,\cdot ⟩:V\times W\to \mathbb{K}$.

(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 $\mathbb{K}$. We define the weak topology on $X$, denoted by $\sigma (X,Y)$, as the initial topology induced by the maps $⟨\cdot ,y⟩:X\to \mathbb{K}$, where $y\in Y$. Similarly, the weak topology on $Y$, denoted by $\sigma (Y,X)$, is the initial topology induced by the maps $⟨x,\cdot ⟩:Y\to \mathbb{K}$, where $x\in X$.

Theorem (Weak Topologies are Locally Convex)

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

The Canonical Pairing

TODO: Def & Theorem (weak rep)

Definition (Polar Set)

Suppose $⟨X,Y⟩$ is a dual pair of vector spaces. The polar of a subset $A\subset X$ is the set

$$A^{\circ }=\{y\in Y:|⟨x,y⟩|\le 1\forall x\in A\}.$$

The polar of a subset $B\subset Y$ is the set

$$B^{\circ }=\{x\in X:|⟨x,y⟩|\le 1\forall y\in B\}.$$

Some authors define the polar with the condition $\mathrm{Re}⟨x,y⟩\le 1$ instead of $|⟨x,y⟩|\le 1$ 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 $X$.

Bipolar Theorem

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

$$A^{\circ \circ }=\overline{\mathrm{aco}(A)},$$

where the closure is taken with respect to the weak topology on $X$, that is $\sigma (X,Y)$.