Proposition 1

Let E be a set, $\mathfrak{S}$ a family of subsets of E, F a topological vector space and H a vector subspace of $F^E$. In order that the $\mathfrak{S}$-topology* be compatible with the vector space structure of H, it is necessary and sufficient that $u(M)$ is bounded in F for every $u \in H$ and every $M \in \mathfrak{S}$. If, moreover, F is locally convex, then the $\mathfrak{S}$*-topology* on H is locally convex.*

On account of Remarks 1) and 2) above, we see that a necessary and sufficient condition for the $\mathfrak{S}$*-topology* to be compatible with the vector space structure of H is that the sets $H \cap T(M, V)$ are absorbent in H (I, p. 7, prop. 4); but this implies that for every $u \in H$, every subset $M \in \mathfrak{S}$, and every balanced neighbourhood V of 0 in F, there exists $\lambda \neq 0$ such that $u(M) \subset \lambda V$; that is to say (III, p. 2) that $u(M)$ is bounded in F. Finally, the last assertion of the proposition follows from the fact that if V is convex, so is $T(M, V)$.