Definition 1

Let $E$ be a commutative topological group with operators, and let $(M_i)_{1 \leq i \leq n}$ be a finite family of stable subgroups of $E$, such that $E$ is the direct sum of the $M_i$. Then $E$ is said to be the topological direct sum of the $M_i$ if the canonical mapping $(x_i) \to \sum_{i=1}^n x_i$ of the product group $\prod_{i=1}^n M_i$ onto $E$ is a homeomorphism (and therefore an isomorphism of topological groups with operators).