Corollary 2

— Let E be a Hausdorff prehilbertian space, $(E_i)_{i \in I}$ a family of complete vector subspaces of E such that, for every pair of distinct indices $i, k$ in I, the subspaces $E_i$ and $E_k$ are orthogonal. Let V be the closed vector subspace of E generated by the union of the $E_i$. For every $i \in I$, let $p_{E_i}$ be the orthoprojector from E onto $E_i$. Let $x \in E$.

1) We have $\sum_{i \in I} \|p_{E_i}(x)\|^2 \leq \|x\|^2$.

2) The following conditions are equivalent : a) $x \in V$; b) $\sum_{i \in I} \|p_{E_i}(x)\|^2 = \|x\|^2$; c) the family $(p_{E_i}(x))_{i \in I}$ is summable in E, and we have $x = \sum_{i \in I} p_{E_i}(x)$.

3) Suppose V is complete. Then the family $(p_{E_i}(x))_{i \in I}$ is summable in E, and

where $p_V$ denotes the orthoprojector from E onto V.

Let $\hat{E}$ be the hilbertian space completion of E; we identify E with a dense subspace of $\hat{E}$; the $E_i$, being complete, are closed subspaces of $\hat{E}$. The closure $\overline{V}$ of V in $\hat{E}$ is the closed vector subspace of $\hat{E}$ generated by the union of the $E_i$, and $V = \overline{V} \cap E$. The space $\hat{E}$ is the hilbertian sum of the $E_i$ and of the subspace W, the orthogonal complement of $\overline{V}$ in $\hat{E}$; put $x_0 = p_W(x)$ and $x_i = p_{E_i}(x)$ for all $i \in I$. By cor. 1, we have $\|x\|^2 = \|x_0\|^2 + \sum_{i \in I} \|x_i\|^2$, and $x = x_0 + \sum_{i \in I} x_i$ in $\hat{E}$. This implies assertion 1), and the fact that conditions b) and c) of 2) are equivalent to the condition $x_0 = 0$, hence to the condition $x \in V$. Finally, if V is complete, and if we put $x' = p_V(x)$, we have $x' - x_i = (x - x_i) - (x - p_V(x))$, hence $x' - x_i$ is orthogonal to $E_i$, and so $x_i = p_{E_i}(x')$ for all $i \in I$; it is now enough to apply property 2) to the vector $x'$.