Theorem 2

*Let E be a prehilbertian space and M a vector subspace of E, which is Hausdorff and complete. Then E is the topological direct sum of M and of $M^\circ$ the subspace orthogonal to M. The projector from E onto M associated with the decomposition $E = M \oplus M^\circ$ is the projection $p_M$ from E onto M defined in th. 1 (V, p. 10).

We first show that $x - p_M(x)$ belongs to $M^\circ$ for all $x \in E$. Let $y \in M$. For every scalar $\lambda \in K$, the vector $p_M(x) + \lambda y$ belongs to M; hence by formula 15 (V, p. 10) we have,

for all $\lambda \in K$. If, in particular we take $\lambda = \overline{\langle x - p_M(x)|y \rangle}$ we conclude that $\langle x - p_M(x)|y \rangle = 0$, hence our assertion.

Since M is Hausdorff, 0 is the only vector of M, orthogonal to itself, hence $M \cap M^\circ = \{0\}$. For every $x \in E$, we have $p_M(x) \in M$ and $x - p_M(x) \in M^\circ$. Consequently, E is the direct sum of M and $M^\circ$, and $p_M$ is the projector from E onto M with kernel $M^\circ$. Since $p_M$ is a continuous mapping from E into M (V, p. 11, cor. 2), if follows from GT, III, § 6, No. 2 that E is the topological direct sum of M and $M^\circ$.