Definition 2

Let F and G be two (real) vector spaces in duality. For every set M of F we call the polar of M, the set of those y ∈ G for which $\langle x, y \rangle \geq -1$ for all $x \in M$. (For complex vector spaces, cf. II, p. 64.)

If $G_1, G_2$ are two subspaces of $F^*$ such that $G_1 \subset G_2$, then the polar of M in $G_1$ is the intersection of $G_1$ with the polar of M in $G_2$.

When there is no danger of confusion we use $M^\circ$ to denote the polar, in G, of the subset M of F. Similarly we define the polar in F of a set in G.

Obviously, for every scalar $\lambda \neq 0$ and all $M \subset F$, we have $(\lambda M)^\circ = \lambda^{-1} M^\circ$. The relation $M \subset N \subset F$ implies $N^\circ \subset M^\circ$; if N absorbs M then $M^\circ$ absorbs $N^\circ$; for every family $(M_\alpha)$ of sets of F, the polar set of $\bigcup \alpha M_\alpha$ is the intersection of the polar sets $M_\alpha^\circ$. Since, for $y \in M^\circ$, the closed half-spaces defined by the relations $\langle x, y \rangle \geq -1$ contain 0 and M, we see that if $M_1$ is the convex envelope of $M \cup \{0\}$, then $M_1^\circ = M^\circ$.

Clearly $M \subset M^{\circ \circ}$. Hence

i.e. $M^{\circ \circ \circ} = M^\circ$ (cf. S, III, § 1.5, prop. 2).

If M is a symmetric subset of F, $M^\circ$ is a symmetric subset of G; $M^\circ$ is also in this case the set of $y \in G$ such that $|\langle x, y \rangle| \leq 1$ for all $x \in M$.