Proposition 1

— Let E be a Hausdorff locally convex space. Suppose that there exists a sequence (E_n)_{n \in \mathbf{N}} of Fréchet spaces satisfying the first axiom of countability, and continuous linear mappings u_n : E_n → E such that E = \bigcup_{n \in \mathbf{N}} u_n(E_n). Then E is a Lusin space.

Let P_n be the kernel of u_n; then u_n defines a bijective continuous mapping from the quotient space E_n/P_n onto u_n(E_n). Since E_n/P_n is a Fréchet space satisfying the first axiom of countability (GT, IX, § 3, No. 1), hence a polish space (GT, IX, § 6, No. 1, def. 1), u_n(E_n) is a Lusin subspace of E (GT, IX, § 6, No. 4, prop. 11). Therefore by GT, IX, § 6, No. 7, cor. of th. 3, the space E, which is regular (GT, III, § 3, No. 1) is a Lusin space.