Remark

— 1) When E is a topological vector space, the dual $E^*$ of the vector space E will be called the algebraic dual of E to avoid confusion. We note also that $E^*$ is the dual of the topological vector space obtained by giving E the finest locally convex topology (II, p. 25, Example 2). 2) The dual E' of a topological vector space does not itself carry a topology, unless this is expressly stated. 3) If F and $G \subset F^*$ are in separating duality by the canonical bilinear form, then this is also true of F and $G_1$, for every subspace $G_1$ of $F^*$ such that $G \subset G_1$.