Proposition 8

Let $N_\iota$ be a subspace of $E_\iota$, for every $\iota \in I$,

(i) The topology induced on $N = \sum_\iota N_\iota$ by the direct sum topology $\mathscr{T}$ on $E$ is identical with the direct sum of the topologies of the $N_\iota$.

(ii) The canonical mapping $h$ of the topological direct sum space of the $E_\iota / N_\iota$ on $E/N$ (A, II, § 1.6, formula (26)) is an isomorphism between topological vector spaces.

(i) With the notations introduced above, we consider $x = \sum_\iota \lambda_\iota x_\iota$ belonging to $N \cap \Gamma((V_\iota))$ ($(\lambda_\iota)$ is a family of numbers $\geqslant 0$ of which at most finitely many are non-zero, such that $\sum_\iota \lambda_\iota = 1$, and $x_\iota \in V_\iota$, for all $\iota \in I$). Since the sum of the $N_\iota$ is direct, we have $\lambda_\iota x_\iota \in N_\iota$ for all $\iota \in I$; therefore, for all $\iota$ such that $\lambda_\iota > 0$ we also have $x_\iota \in N_\iota \cap V_\iota$, and $x$ belongs to the convex envelope $\Gamma((N_\iota \cap V_\iota))$, thus (i) is proved.

(ii) Denote canonical mappings as follows: $f_\iota : E_\iota \to E, h_\iota : E_\iota / N_\iota \to E/N, p_\iota : E_\iota \to E_\iota / N_\iota$ and $p : E \to E/N$. For every $\iota \in I$, $h_\iota \circ p_\iota = p \circ f_\iota$ and the proposition follows from II, p. 28, cor. 2 and p. 29 Example I.