Theorem 1

— Let E be a locally convex space which is the inductive limit of Banach spaces, F a Souslin locally convex space, for example a Lusin space (GT, IX, § 6, No. 2 and No. 4), and u a linear mapping from E into F. If the graph of u is a Borel subset of E × F, then u is continuous.

Let E_i be a family of Banach spaces, and (u_i) a family of continuous linear mappings u_i : E_i → E such that the topology of E is the finest locally convex topology for which the u_i are continuous. It is enough to prove that the composed mappings u ∘ u_i are continuous, or in fact (GT, IX, § 2, No. 6, prop. 10) that the restriction of u ∘ u_i to every closed subspace G of E_i satisfying the first axiom of countability is continuous. The graph of this restriction is the inverse image of the graph of u under the continuous mapping u_i × Id_F : G × F → E × F, hence is a Borel set in G × F. In addition, G × F is a Souslin space and every Borel subset of a Souslin space is a Souslin space (GT, IX, § 6, No. 3, prop. 10). Th. 1 then follows from th. 4, GT, IX, § 6, No. 8.