Definition 1

Let $F$ and $G$ be two complex vector spaces in duality. For any subset $M$ of $F$, the polar of $M$ in $G$, denoted by $M^\circ$, is the set of $y \in G$ such that $\Re(\langle x, y \rangle) \geq -1$ for all $x \in M$.

If $M^\circ$ is the polar of $M \subset F$ in $G$ then $(\lambda M)^\circ = \lambda^{-1}M^\circ$ for all $\lambda \in \mathbf{C}^*$.

If $M$ is a (complex) vector subspace of $E$, then $M^\circ$ is a closed vector subspace (for $\sigma(G, F)$), since the relation $\Re(\lambda \langle x, y \rangle) \geq -1$ for every scalar $\lambda \in \mathbf{C}$ implies $\langle x, y \rangle = 0$; again we say that $M^\circ$ is the subspace of $G$ orthogonal to $M$.

If $M$ is a balanced set in $F$, then $M^\circ$ is a balanced set in $G$; in this case $M^\circ$ is the set of $y \in G$ such that $|\langle x, y \rangle| \leq 1$ for all $x \in M$; for this relation is equivalent to $\Re(\langle \zeta x, y \rangle) \leq 1$ for all $x \in M$ and all $\zeta \in \mathbf{C}$ such that $|\zeta| = 1$.

The results of II, p. 41 to 51 are also valid without restriction for complex vector spaces.

Exercises