Bipolar theorems

Let $\langle V,W\rangle$ be a duality of complex vector spaces and $K$ a set of matrices over $V$.

The matrix polar of $K$ is a set $D$ of matrices over $W$, given by⁶⁴

$$D_n=\{w\in M_n(W)\;\vert\;\mathrm{Re}\langle v,w\rangle\leqslant\mathrm{1\!\!\!\:l}_{nm}$$   $$\mbox{ for all $m\in{\mathbb{N}}$, $v\in K_m$}$$$$\}$$.

The absolute matrix polar of $K$ is a set $D$ of matrices over $W$, given by⁶⁵

$$D_n=\{w\in M_n(W)\;\vert\;\Vert\langle v,w\rangle\Vert\leqslant 1$$   $$\mbox{ for all $m\in{\mathbb{N}}$, $v\in K_m$}$$$$\}$$.

Polars of sets of matrices over $W$ are defined analogously.

We have the bipolar theorems: Let $\langle V,W\rangle$ be a duality of complex vector spaces and $K$ a set of matrices over $V$.

a)

[EW97b] $K$ equals its matrix bipolar if and only if $K$ is closed and matrix convex and $0\in K_1$.

b)

[EW97a] $K$ equals its absolute matrix bipolar if and only if $K$ is closed and absolutely matrix convex.

The matrix bipolar of a set $K$ of matrices over $V$ is therefore the smallest closed and matrix convex set which contains $K$ and 0.

The absolute matrix bipolar of a set $K$ of matrices over $V$ is therefore the smallest closed and absolutely matrix convex set which contains $K$.

So we get a characterization of the unit balls of $\mathit{MIN}(E)$ and $\mathit{MAX}(E)$ for a normed space $E$.

Footnotes

... by⁶⁴

$D=\{w\in M(W)\;\vert\;Re\langle v,w\rangle\leqslant\mathrm{1\!\!\!\:l}$    for all $v\in K$$\}$.

... by⁶⁵

$D=\{w\in M(W)\;\vert\;\Vert\langle v,w\rangle\Vert\leqslant 1$    for all $v\in K$$\}$.