Theorem 2 (Šmulian)

— Let E be a Fréchet space and A a subset of E. Let $E_\sigma$ denote the space E endowed with the weakened topology. The following conditions are equivalent :

(i) A is relatively compact in $E_\sigma$; (ii) every infinite sequence of points of A has a limit point in $E_\sigma$; (iii) from every infinite sequence of points of A, we can extract a sequence which converges in $E_\sigma$.

The equivalence of (i) and (ii) follows from Eberlein’s theorem and (iii) obviously implies (ii).

We shall prove that (i) implies (iii). Suppose that the closure B of A in $E_\sigma$ is compact and that $(x_n)_{n \in \mathbf{N}}$ is a sequence of points of A. Let F denote the smallest closed vector subspace of E containing the $x_n$, this is a Fréchet space satisfying the first axiom of countability. Since F is closed in $E_\sigma$ and the topology $\sigma(F, F')$ on F is induced by $\sigma(E, E')$, the set $B \cap F$ is compact for $\sigma(F, F')$. On account of the remarks in IV, p. 32, the existence of a sequence extracted from $(x_n)_{n \in \mathbf{N}}$ converging for $\sigma(E, E')$ (or, which is the same, for $\sigma(F, F')$) is a consequence of the following lemma :