Corollary 3

— Let E be a vector space, (Fi)_{i \in I} a family of locally convex spaces and f_i a linear mapping from E into F_i (for i \in I). Suppose E is assigned the coarsest topology (locally convex) for which all the f_i are continuous (II, p. 26). Then, for a subset A of E to be bounded, it is necessary and sufficient that f_i(A) is bounded in F_i for all i \in I.*

In fact, if A is bounded, so are the f_i(A) (cor. 1). Conversely, if the f_i(A) are bounded and if p is a continuous semi-norm on E, then there exists a finite subset J of I and a family $(q_j)_{j \in J}$, where $q_j$ is a continuous semi-norm on $F_j$, such that $p \leq \sup_{j \in J} (q_j \circ f_j)$ consequently $p$ is bounded on $A$.