Corollary 5 (The closed graph theorem)

— Let E and F be two complete metrisable vector spaces over a non-discrete valued division ring. In order that a linear mapping u of E in F be continuous, it is necessary and sufficient that its graph, in the product space $E \times F$, be closed.

The condition is necessary since the graph of a continuous mapping into a Hausdorff space is closed (GT, I, § 8.1, cor. 2). To see that it is sufficient, note that it implies that the graph G of $u$, which is a closed vector subspace of the complete metrisable space $E \times F$, is itself metrisable and complete. The projection $z \mapsto \mathrm{pr}_1(z)$ of G on E is a bijective, continuous linear mapping, therefore an isomorphism (cor. 1); since its inverse mapping is $x \mapsto (x, u(x))$, it follows that $u$ is continuous in E.

We can express this corollary in the following form : $u$ is continuous if the following situation holds : if the sequence $(x_n)$ of points of E both converges to 0 and is such that the sequence $(u(x_n))$ converges to $y$, then it is necessarily the case that $y = 0$.