Definition 1

— By the dual of a topological vector space E, we mean the subspace E' of E, the dual of the vector space E, formed by the continuous linear forms on E.*

When E is a Hausdorff locally convex space, the duality between E and its dual E' is separating : this follows from the Hahn-Banach theorem (II, p. 24, cor. 1) that for every $x \neq 0$ in E, there exists $x' \in E'$ such that $\langle x, x' \rangle \neq 0$.