(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).

Proposition 1. — The following conditions are equivalent : (i) Every semi-norm on E which is bounded on bounded subsets of E is continuous. (i') Every convex balanced subset of E which absorbs the bounded subsets of E (I, p. 7, def. 4) is a neighbourhood of 0 in E. (ii) E is the inductive limit of the semi-normed spaces $E_A$, where A ranges over the directed increasing set of closed, convex, balanced and bounded subsets of E. (ii') There exists a family $(E_i)_{i \in I}$ of semi-normed spaces, and for every $i \in I$, a linear mapping $u_i : E_i \to E$ such that the topology of E is the finest locally convex topology for which the $u_i$ are continuous. (iii) For an arbitrary locally convex space F, a linear mapping $u : E \to F$ is continuous if and only if for every sequence $(x_n)$ of points in E tending to 0, the sequence $(u(x_n))$ is bounded in F.

(iii') For an arbitrary semi-normed space F, a linear mapping u : E → F is continuous if and only if u(X) is bounded in F for every bounded set X in E.

It is immediate that (i) and (i') are equivalent in view of the correspondence between semi-norms and convex, balanced, absorbent subsets (II, p. 20). If p is a semi-norm on E, which is continuous on each E_A, then p is bounded on bounded subsets of E; hence (i) implies (ii) (II, p. 27, prop. 5). It is clear that (ii) implies (ii').

Now let (E_i, u_i)_{i \in I} be as in (ii') and let u be a linear mapping from E into a locally convex space F, such that (u(x_n)) is bounded in F for every sequence (x_n) of points of E tending to 0. It follows from lemma 1 of III, p. 11 that the linear mapping u \circ u_i : E_i \to F is continuous for all i \in I. Hence, if the topology of E is the finest locally convex topology for which the u_i are continuous, then u is continuous (II, p. 27, prop. 5). This proves that (ii') implies (iii).

It is immediate that (iii) implies (iii') (III, p. 3, cor.) Finally, if p is a semi-norm on E, which is bounded on bounded subsets of E, the condition (iii') asserts that the identity map is continuous from E into the semi-normed space (E, p); in other words, p is continuous. This proves that (iii') implies (i).

In fact, the spaces E_A, where A is closed and bounded are Banach spaces (III, p. 8, corollary).

Suppose E is metrizable, and p a semi-norm on E which is bounded on bounded subsets of E, but which is not continuous. Let A be the set of all $x \in E$ such that $p(x) < 1$. Let $(V_n)_{n \geq 1}$ be a decreasing sequence forming a fundamental system of neighbourhoods of 0 in E. Since p is not continuous, A is not a neighbourhood of 0; hence for every $n > 0$, we have $A \not\supseteq n^{-1} V_n$ and there exists a point $x_n$ in $V_n$, such that $n^{-1} x_n \notin A$, that is, $p(x_n) \geq n$. The sequence $(x_n)$ tends to 0, hence is bounded (III, p. 3, corollary); this contradicts the hypothesis on p.