Proposition 5

— Let E and F be two locally convex spaces and let T be a total subset of E. The following uniform structures coincide on every equicontinuous subset H of $\mathscr{L}(E; F)$:

1) the uniform structure of simple convergence in T; 2) the uniform structure of simple convergence in E; 3) the uniform structure of convergence in the precompact subsets of E.

We recall (III, p. 15, prop. 2) that the $\mathfrak{S}$-topology on $\mathscr{L}(E; F)$ coincides with the $\tilde{\mathfrak{S}}$-topology, where $\tilde{\mathfrak{S}}$ is the smallest bornology adapted to E and containing $\mathfrak{S}$.

In the statement of prop. 5, we can therefore replace the word « total » by « everywhere dense ». The proposition then follows from the general properties of equicontinuous sets (GT, X, § 2, No. 4, th. 1).