Proposition 1

Let E, F be two topological vector spaces over a non-discrete valued division ring K. We suppose that :

1) E is metrisable and complete. 2) There exists a sequence $(F_n)$ of complete metrisable vector spaces over K and, for each n, an injective continuous linear mapping $v_n$ of $F_n$ in F such that F is the union of the subspaces $v_n(F_n)$.

Then let u be a linear mapping of E in F. If the graph of u is closed in $E \times F$, then there exists an integer n and a continuous linear mapping $u_n$ of E in $F_n$ such that $u = v_n \circ u_n$ (which implies that u is continuous and that $u(E) \subset v_n(F_n)$).*

Let G be the graph of $u$ in $E \times F$. For all $n$, we consider the continuous linear mapping $w_n : (x, y) \mapsto (x, v_n(y))$ of $E \times F_n$ in $E \times F$; as G is closed, the set $w_n^{-1}(G) = G_n$ is a closed vector subspace of $E \times F_n$; if $p_n$ is the restriction to $G_n$ of the first projection $\mathrm{pr}_1$, we have $p_n(G_n) = u^{-1}(v_n(F_n))$. As $p_n$ is continuous and $G_n$ is complete (since $G_n$ is closed in the complete space $E \times F_n$), $p_n(G_n)$ is, by theorem 1, either meagre in E or it is the whole of E. But, by hypothesis, E is the union of the $p_n(G_n)$, and as E is complete, the $p_n(G_n)$ cannot all be meagre in E by Baire’s theorem (GT, IX, § 5.3, th. 1). Therefore there exists an integer n such that $p_n(G_n) = E$, or in other words $u(E) \subset v_n(F_n)$. Further, as $v_n$ is injective, $G_n$ is the graph of a linear mapping $u_n$ of E in $F_n$, and by the closed graph theorem (I, p. 19, cor. 5) $u_n$ is continuous; it follows then from the definitions that $u = v_n \circ u_n$.

Exercises