Lemma 1

— Let F be a Fréchet space satisfying the first axiom of countability. Every subset C of F which is compact for the topology $\mathscr{T}$ induced by $\sigma(F, F')$ is metrizable for this topology.

Since the topology of precompact convergence on $F'$ is finer than the topology $\sigma(F', F)$, there exists an everywhere dense countable subset in $F'_s$ (III, p. 18, cor. 1). Hence the set C can be identified with a subset of $K^D$, and the topology induced on C by that of $K^D$, which is metrizable (GT, IX, § 2, No. 8) is coarser than the topology induced by $\sigma(F, F')$, for which C is compact. Hence these two topologies are identical (GT, I, § 9, No. 4, cor. 3).

Šmulian’s theorem can be extended to the case where E is the strict inductive limit of a sequence of Fréchet spaces (IV, p. 67, exerc. 2).