Proposition 1

Let T be a subset of E. The following conditions are equivalent:

(i) T is convex, balanced, closed and absorbent.

(ii) T is the polar of a convex, balanced and weakly bounded set M in E'.

(iii) There exists a lower semi-continuous semi-norm p on E, such that T is the set of all x ∈ E satisfying p(x) ≤ 1.

(i) ⇒ (ii) : under the hypothesis of (i), let M = T°; then M is convex and balanced in E'. For every x ∈ E, there exists a real number r > 0 such that rx ∈ T, hence |u(x)| ≤ $\frac{1}{r}$ for all u ∈ M; in other words M is weakly bounded. From cor. 3 of II, p. 45, we have T = M°, hence T satisfies (ii).

(ii) ⇒ (iii) : under the hypothesis of (ii), let $p(x) = \sup_{u \in M} |u(x)|$ for all x ∈ E. It is immediate that T = M° consists of all x ∈ E such that p(x) ≤ 1. The semi-norm p on E' is lower semi-continuous, being the superior envelope of a family of continuous functions (GT, IV, § 6, No. 2, corollary).

(iii) ⇒ (i) : this is clear.