Proposition 6

A locally convex topology $\mathscr{T}$ on $E$ is the direct sum of the topologies of the $E_i$, if and only if, the following property holds: a linear mapping of $E$ in a locally convex space $G$ (resp. a semi-norm $p$ on $E$) is continuous, if and only if, for every $i \in I$, the mapping $g \circ f_i$ (resp. $p \circ f_i$) is continuous in $E_i$.

This is a particular case of prop. 5, II, p. 27.

Recalling the definition of the direct sum of a family of vector spaces (A, II, p. 12, prop. 6), we can say that the topology $\mathscr{T}$ is the only one for which the canonical mapping $g \mapsto (g \circ f_i)$ is a bijection

for every locally convex space $G$.