Proposition 7

— Let $(E_n)_{n \geq 0}$ be a sequence of Hausdorff locally convex spaces, and for every $n$, let $u_n : E_n \to E_{n+1}$ be an injective linear mapping which is compact (i.e. such that there exists a neighbourhood of 0 in $E_n$ whose image under $u_n$ is relatively compact; this implies that $u_n$ is continuous). Let $E$ be the inductive limit of the system $(E_n, u_n)$ (II, p. 29), and let $v_n$ be the canonical mapping from $E_n$ into $E$. Then the locally convex space $E$ is Hausdorff. Moreover, for every subset $A$ of $E$, the following conditions are equivalent : (i) $A$ is bounded; (ii) there exists an integer $n$ such that $A$ is the image under $v_n$ of a bounded subset of $E_n$; (iii) $A$ is relatively compact. We identify $E_n$ with a vector subspace of $E$ (endowed with a topology finer than the induced topology).