Proposition 3

— Let $F$ and $G$ be two real vector spaces in duality. Every linear form on $F$, that is continuous for $\sigma(F, G)$, can be written as $x \mapsto \langle x, y \rangle$ for some $y \in G$. The element $y \in G$ is unique when the duality is separating in $G$.

For, to say that the linear form $f$ on $F$ is continuous for $\sigma(F, G)$ means that there exists a finite set of points $y_i \in G$ ($1 \leq i \leq n$) such that, for all $x$ in $F$, $|f(x)| \leq \sup_{1 \leq i \leq n} |\langle x, y_i \rangle|$ (II, p. 6, prop. 5). The $n$ relations $\langle x, y_i \rangle = 0$ ($1 \leq i \leq n$) imply therefore $f(x) = 0$, and hence (A, II, § 7.5, cor. 1), there exists a linear combination $y = \sum_{i=1}^n \lambda_i y_i$ such that $f(x) = \langle x, y \rangle$ for all $x \in F$. The uniqueness follows from (D$_\text{II}$).

In other words, when the duality is separating in $G$, and $F$ has the topology $\sigma(F, G)$, then we can identify $G$ canonically with the dual of $F$ for this topology (II, p. 42, def. 1).