Proposition 1

(i) The closed convex subsets in E are the same for all the locally convex topologies on E which are compatible with the duality between E and F. (ii) The bounded subsets of E are the same for all the locally convex topologies on E which are compatible with the duality between E and F.

Let $\mathscr{T}$ be a topology on E compatible with the duality between E and F, hence finer than $\sigma(E, F)$. If a convex subset of E is closed for $\mathscr{T}$, it is the intersection of closed, real half-spaces (II, p. 38, cor. 1), hence it is closed for $\sigma(E, F)$. This proves (i). Assertion (ii) was proved in cor. 3 of III, p. 27.

Let $F_\sigma$ denote the vector space F endowed with the weak topology $\sigma(F, E)$. Then the linear mapping $s_B$ maps E onto the dual $(F_\sigma)'$ of $F_\sigma$ (II, p. 43, prop. 3). Let $\mathfrak{S}$ be a family of bounded subsets of $F_\sigma$. By abuse of language, the inverse image under $s_B$ of the $\mathfrak{S}$-topology on $(F_\sigma)'$ is called the $\mathfrak{S}\text{-topology on } E$. It is defined by the family of semi-norms

where A runs through $\mathfrak{S}$. In particular, when $\mathfrak{S}$ is the family of finite subsets of F, the $\mathfrak{S}$-topology is precisely the weak topology $\sigma(E, F)$.