Proposition 9

Let $R, S, T$ be three locally convex Hausdorff spaces. Suppose that the spaces $\mathscr{L}(R; S), \mathscr{L}(S; T), \mathscr{L}(R; T)$ are each assigned the topology of simple (resp. compact, bounded) convergence. Then the bilinear mapping $(u, v) \mapsto v \circ u$ from $\mathscr{L}(R; S) \times \mathscr{L}(S; T)$ into $\mathscr{L}(R; T)$ is $(\mathfrak{S}, \mathfrak{T})$-*hypocontinuous*, where $\mathfrak{T}$ is the family of equicontinuous subsets of $\mathscr{L}(S; T)$, and $\mathfrak{S}$ the family of finite (resp. compact, bounded) subsets of $\mathscr{L}(R; S)$.

We first prove that $(u, v) \mapsto v \circ u$ is $\mathfrak{T}$-hypocontinuous. Let H be an equicontinuous set in $\mathscr{L}(S; T)$, let W be a neighbourhood of 0 in T and let M be a finite (resp. compact, bounded) subset of R. We must show that there exists a neighbourhood V of 0 in S such that if $u(M) \subset V$ and $v \in H$, then $v(u(M)) \subset W$. But for this, it is enough to have $v(V) \subset W$ for all $v \in H$, and the existence of such a neighbourhood follows from the equicontinuity of H.

To see that $(u, v) \mapsto v \circ u$ is $\mathfrak{S}$-hypocontinuous, we shall prove that, for every neighbourhood W of 0 in T, every finite (resp. compact, bounded) subset M of R and every finite (resp. compact, bounded) subset L of $\mathscr{L}(R; S)$ there exists a finite (resp. compact, bounded) subset N of S such that the relations $v(N) \subset W$ and $u \in L$ imply that $v(u(M)) \subset W$. Evidently it is enough to show that we can take $N = \bigcup_{u \in L} u(M)$, i.e. that the set N is finite (resp. compact, bounded) whenever L and M are. This is immediate if L and M are finite, or if M is bounded in R and L is bounded in $\mathscr{L}(R; S)$ (for the topology of bounded convergence, cf. III, p. 22). Finally, we show that if M is compact in R and L is compact in $\mathscr{L}(R; S)$ for the topology of compact convergence, then N is compact in S. But if $u_M$ is the restriction to M of $u \in L$, the mapping $u \mapsto u_M$ from L into the space $\mathscr{C}(M; S)$ of all continuous mappings from M into S, with the topology of uniform convergence, is continuous; hence the image of L under this mapping is compact, and our assertion then follows from the continuity of the map $(w, x) \mapsto w(x)$ from $\mathscr{C}(M; S) \times M$ into S (GT, X, § 1, No. 6, prop. 9).

In the two corollaries that follow, we assume as in prop. 9, that the spaces $\mathscr{L}(R; S)$, $\mathscr{L}(S; T)$, $\mathscr{L}(R; T)$ are all three assigned the topology of simple convergence, or all three the topology of compact convergence, or all three that of bounded convergence.