Theorem 2

— Let E be a locally convex space, which is the inductive limit of an increasing sequence $(E_n)_{n \in \mathbf{N}}$ of subspaces of E, endowed with the topologies of Fréchet spaces satisfying the first axiom of countability. Suppose that every compact subset of E is contained in one of the $E_n$ and is compact in this space. Let F be a Fréchet space satisfying the first axiom of countability. Then the space $\mathscr{L}_c(E; F)$ is a Lusin space.

The space E is bornological (III, p. 12), hence the space $\mathscr{L}_c(E; F)$ is complete (III, p. 23, prop. 12). The linear mapping $j : f \mapsto (f|E_n)_{n \in \mathbf{N}}$ is an injection from $\mathscr{L}_c(E; F)$ into the product space $\prod_{n \in \mathbf{N}} \mathscr{L}_c(E_n; F)$; by virtue of the hypothesis on the compact subsets of E and the definition of the $\mathfrak{S}$-topologies, $j$ is an isomorphism from $\mathscr{L}_c(E; F)$ onto its image (endowed with the topology induced by the product topology); moreover, since $\mathscr{L}_c(E; F)$ is complete, this image is a closed subspace of $\prod_{n \in \mathbf{N}} \mathscr{L}_c(E_n; F)$ (GT, II, § 3, No. 4, prop. 8). By GT, IX, § 6, No. 4, it is therefore enough to prove that each of the spaces $\mathscr{L}_c(E; F)$ is a Lusin space. For the rest of the proof, we shall assume that E is a Fréchet space satisfying the first axiom of countability.

Since F is a Fréchet space satisfying the first axiom of countability, it is isomorphic to a closed subspace of a countable product of Banach spaces $F_n$, each of which is a quotient of F (II, p. 5), hence satisfies the first axiom of countability. The linear mapping $j' : f \mapsto (\mathrm{pr}_n \circ f)_{n \in \mathbf{N}}$ is an injection from $\mathscr{L}_c(E; F)$ into the product space $\prod_{n \in \mathbf{N}} \mathscr{L}_c(E; F_n)$, and by using the definition of the $\mathfrak{S}$-topologies and of the open sets in a product, $j'$ is an isomorphism from $\mathscr{L}_c(E; F)$ onto its image; moreover, since $\mathscr{L}_c(E; F)$ is complete, this image is a closed subspace of $\prod_{n \in \mathbf{N}} \mathscr{L}_c(E; F)$. Therefore it is enough to prove that each of the spaces $\mathscr{L}_c(E; F_n)$ is a Lusin space (GT, IX, § 6, No. 4), and consequently, we can assume that F is a Banach space satisfying the first axiom of countability.

The space $\mathscr{L}_c(E; F)$ is the union of a countable family of equicontinuous and closed subsets (III, p. 19, cor. 1 and GT, X, § 2, No. 3, prop. 6). But every equicontinuous subset H of $\mathscr{L}_c(E; F)$ is metrizable and satisfies the first axiom of countability (III, p. 18, prop. 6 and GT, X, § 2, No. 4, th. 1); if H is closed, then it is a complete space for the uniform structure induced by that of $\mathscr{L}_c(E; F)$, since the latter is complete In other words, H is a polish space, and a fortiori a Lusin space; consequently the regular space $\mathscr{L}_c(E; F)$ is a Lusin space (GT, IX, § 6, No. 7, cor. of th. 3).