Proposition 2

Let $E$ be a commutative topological group with operators which is the direct sum of stable subgroups $M_i$ ($1 \leq i \leq n$). Let $(p_i)_{1 \leq i \leq n}$ be the family of projectors associated with the decomposition $E = \sum_{i=1}^n M_i$. Then $E$ is the topological direct sum of the $M_i$ if and only if the $p_i$ are continuous.

For the mapping $x \to (p_i(x))$ is the inverse of the mapping $(x_i) \to \sum_{i=1}^n x_i$.

Since $1 = \sum_{i=1}^n P_i$ (where $1$ denotes the identity mapping of $E$) it is sufficient for $n - 1$ of the projectors $p_i$ to be continuous in order that the $n$ th should also be continuous.

If $E$ is the topological direct sum of two stable subgroups $M, N$, then $N$ is said to be a topological complement of $M$ in $E$; this is the case if and only if the canonical mapping of $E/M$ onto is an isomorphism of topological groups with operators.