Exercise 5, § 3

Let E and F be two complete metrisable vector spaces over a non-discrete valued division ring and let $u$ be a linear mapping of E in F. Let N be the set of cluster points of $u$ in F with respect to the filter of neighbourhoods of 0 in E; show that N is a closed vector subspace of F, and that, in order that $u$ be continuous, it is necessary and sufficient that N be the single point 0 (use I, p. 19, cor. 5). Show that N is the smallest of the closed vector subspaces M of F such that, if $\phi$ denotes the canonical homomorphism of F on F/M, then $\phi \circ u$ is a continuous mapping of E in F/M.