Corollary

*Let E be a Hausdorff prehilbertian space and M a finite dimensional vector subspace of E. Then E is the direct sum of M and $M^\circ$.

Since E is Hausdorff, so is M; since M is finite dimensional, it is complete (I, p. 13). It is therefore enough to apply th. 2.

With the notations of th. 2, we say that $M^\circ$ is the orthogonal complement of M and that $p_M$ is the orthoprojector (or the orthogonal projector, or by abuse of language, the projector) from E onto M; if x is a vector of E, the vector $p_M(x)$ of M is also called the orthogonal projection of x on M. Note that $p_M$ is a continuous linear mapping from E onto M and that we have $\|p_M\| = 1$ by cor. 2 of V, p. 11, except in the case when $M = \{0\}$ in which case $p_M = 0$.

It follows immediately from Pythagoras theorem that the canonical mapping $\psi$ from $E/M$ onto $M^\circ$ deduced from the direct sum decomposition $E = M \oplus M^\circ$ is isometric if $E/M$ is assigned the quotient semi-norm from that of E (II, p. 4). We shall always assign that prehilbertian structure to $E/M$ for which $\psi$ is an isomorphism of prehilbertian spaces; the quotient semi-norm on $E/M$ is then deduced from this prehilbertian structure.

We shall often use the preceding results when E is a hilbertian space and M a closed vector subspace of E. In this case, $M^\circ$ is a closed vector subspace of E, and $p_{M^\circ} = 1 - p_M$, and $(M^\circ)^\circ = M$.