Definition 2

Let E be the vector space which is the direct sum (A, II, § 1.6) of the family of locally convex spaces $(E_i)_{i \in I}$. For each $i \in I$, let $f_i$ be the canonical injection of $E_i$ in $E$. By the topological direct sum of the family $(E_i)$ we mean the space $E$ with the finest locally convex topology which makes each $f_i$ continuous (this topology is called the direct sum of the topologies of the $E_i$).

In the remainder of this No. we keep the same notations as in def. 2 (unless the contrary is expressly stated) and we identify, canonically, each $E_i$ with a subspace of $E$, by means of $f_i$.

By the general description of neighbourhoods of a locally convex final topology given in II, p. 28, we can here obtain a fundamental system of neighbourhoods of 0 in $E$ for the direct sum topology, in the following manner; for every family $(V_i)_{i \in I}$ where $V_i$ is a symmetric convex neighbourhood of 0 in $E_i$, consider the convex envelope $\Gamma((V_i))$, of the union of the $V_i$; the $\Gamma((V_i))$ for all the families $(V_i)$ (or only taking $V_i$ for each $i$ to be in a fundamental system of neighbourhoods of 0 in $E_i$) form a fundamental system of neighbourhoods of 0 in $E$.