Corollary 1

— Let E be a hilbertian space, which is a hilbertian sum of a family $(E_i)_{i \in I}$ of closed vector subspaces of E; for all $i \in I$, let $p_{E_i}$ be the orthoprojector (V, p. 13) from E onto $E_i$.

a) For all $x \in E$, the family $(\|p_{E_i}(x)\|^2)_{i \in I}$ is summable in $\mathbf{R}$, the family $(p_{E_i}(x))_{i \in I}$ is summable in E, and we have

b) Conversely, if $(x_i)_{i \in I}$ is a family of elements of E such that $x_i \in E_i$ for all $i \in I$ and $\sum_{i \in I} \|x_i\|^2 < + \infty$, this family is summable, and the sum $x$ is the only point of E for which $p_{E_i}(x) = x_i$ for all $i \in I$.

c) For every pair of points $x, y$ of E, we have

These properties are in fact obvious for the external hilbertian sum of the $E_i$, and can be transferred to E by isomorphism.