Definition 2

— Let F and G be two vector spaces put in duality by the bilinear form B. The coarsest topology on F that makes all the linear forms $B(., y): x \mapsto B(x, y)$ continuous, where y varies in G, is called the weak topology on F defined by the duality between F and G, and we denote it by $\sigma(F, G)$.

Similarly we define the weak topology $\sigma(G, F)$ on G, interchanging F and G in definition 1 ; this possibility of interchanging F and G applies to all the results and definitions that follow in this paragraph.

We use the adjective « weak » and the adverb « weakly » to denote properties relative to a weak topology $\sigma(F, G)$ provided there is no possibility of confusion. We shall speak, for example, of « weak convergence » and « weakly continuous functions » etc.

When $G \subset F^*$, the notation $\sigma(F, G)$ will always denote the weak topology defined by the duality corresponding to the restriction to $F \times G$ of the canonical bilinear form $(x, x^*) \mapsto \langle x, x^* \rangle$.

Without extra hypotheses on F and G, we often write $\langle x, y \rangle$ for the value $B(x, y)$ of the bilinear form B at $(x, y)$, provided there is no ambiguity ; we shall adopt this convention in the rest of this paragraph.

A vector space F carrying a weak topology of $\sigma(F, G)$ will be called a weak space.

A weak topology $\sigma(F, G)$ is locally convex (II, p. 26, prop. 4) ; more precisely, it is the inverse image of the product topology of $\mathbf{R}^G$ by the linear mapping $\phi : x \mapsto (\langle x, y \rangle)_{y \in G}$ of F in $\mathbf{R}^G$. It is defined by the set of semi-norms $x \mapsto |\langle x, y \rangle|$ when y varies in G (II, p. 5). For every $\alpha > 0$, and every finite family $(y_i)_{1 \leq i \leq n}$ of points of G, let $W(y_1, ..., y_n; \alpha)$ be the set of the $x \in F$ such that $|\langle x, y_i \rangle| \leq \alpha$ for $1 \leq i \leq n$; these sets (for $\alpha, n$ and $y_i$ arbitrary) form a fundamental system of neighbourhoods of 0 for $\sigma(F, G)$. Note that $W(y_1, ..., y_n; \alpha)$ contains that vector subspace of F, of finite codimension, which is defined by the equations $\langle x, y_i \rangle = 0$ for $1 \leq i \leq n$.