Corollary 1

— Let F and G be two vector spaces in separating duality. If the space F (with $\sigma(F, G)$) is the direct topological sum of two subspaces M, N then the space G (with $\sigma(G, F)$) is the direct topological sum of the subspaces $M^\circ, N^\circ$ orthogonal respectively to M and N.

For let $p : F \to M, q : F \to N$ be the projectors corresponding to the decomposition of F into the direct sum of M and N; in these conditions the mapping $(p, q) : F \to M \times N$ is a topological isomorphism. If $M_1 = G/M^\circ, N_1 = G/N^\circ$, then the topologies on M and N (induced by that of F) are identical with $\sigma(M, M_1)$, $\sigma(N, N_1)$ respectively (II, p. 48, prop. 7). The mapping $^t(p, q) : M_1 \times N_1 \to G$ is a topological isomorphism when we give $M_1, N_1$ and G the topologies $\sigma(M_1, M)$, $\sigma(N_1, N)$ and $\sigma(G, F)$, by prop. 8. Under this mapping $M_1$ (resp. $N_1$) has as its image in G the subspace $N^\circ$ (resp. $M^\circ$), and the topology $\sigma(M_1, M)$ (resp. $\sigma(N_1, N)$) has as its image the topology induced on $N^\circ$ (resp. $M^\circ$) by $\sigma(G, F)$, from which the corollary follows.