Lemma 1

— Let G be a semi-normed space, p its semi-norm, and u a linear mapping from G into E. The following conditions are equivalent :

(i) u is continuous ; (ii) the image of the unit ball of G under u is bounded in E ; (iii) for every sequence $(x_n)$ of points of G tending to 0, the sequence $(u(x_n))$ is bounded in E.

It is immediate that (i) implies (ii) (III, p. 4, cor. 1) and that (ii) implies (iii). Let V be a neighbourhood of 0 in E ; if $u^{-1}(V)$ is not a neighbourhood of 0 in G, then there exists a sequence $(y_n)$ of points of $G - u^{-1}(V)$ such that $p(y_n) \leq \frac{1}{n^2}$. Hence the sequence $x_n = n y_n$ tends to 0 in G and $u(x_n) \notin nV$, which implies that the sequence $(u(x_n))$ is not bounded. Therefore (iii) implies (i).