A statement on prehilbert spaces

Question

Let $(X , \langle \cdot , \cdot \rangle)$ be a prehilbert space and let $M \subset X$ be a vector subspace such that $(M , {\langle \cdot , \cdot \rangle}_M)$ is a Hilbert space, being $$ {\langle \cdot , \cdot \rangle}_M = {\langle \cdot , \cdot \rangle}\big|_{M \times M} : M \times M \to \mathbb{R} \mbox{ (or } \mathbb{C} \mbox{)}\mbox{.} $$ I have to show that $(X , \langle \cdot , \cdot \rangle)$ is a Hilbert space if and only if $\left(M^{\perp} , {\langle \cdot , \cdot \rangle}_{M^{\perp}}\right)$ is a Hilbert space, being $$ {\langle \cdot , \cdot \rangle}_{M^{\perp}} = {\langle \cdot , \cdot \rangle}\big|_{M^{\perp} \times M^{\perp}} : M^{\perp} \times M^{\perp} \to \mathbb{R} \mbox{ (or } \mathbb{C} \mbox{)}\mbox{.} $$

Answer 1

If $X$ is a Hilbert space, then $M^\perp$ as well, as $M^\perp$ is closed.

Assume $M^\perp$ to be a Hilbert space. Let $P_M$ and $P_{M^\perp}$ denote the orthogonal projections onto $M$ and $M^\perp$, respectively. Let $(x_n)$ be a Cauchy sequence in $X$. Then $(P_MX_n)$ and $(P_{M^\perp}x_n)$ are Cauchy sequences and hence convergent. Then also $x_n = P_MX_n + P_{M^\perp}x_n$ is convergent.