Projections on a Hilbert Space over a vector subspace

Question

Im trying to prove the following stament "Let $X$ be a Hilbert space and $M \subseteq X$ a closed vector subspace. For $x \in X$, we have that $$\bar{x} = P_M(x) \iff \bar{x} \in M\ \wedge\ \langle x-\bar{x},z\rangle = 0,\ \ \forall z \in M $$ Where $P_M(x)$ is the unique projection of $x$ on $M$ (i.e the minimun distance to all $z \in M$). $\\$ The stament is obvious $x - \bar{x}$ is parallel to $z \in M$, but in my course we dont, yet, introduce the concept of parallel or perpendicular. Thou, i have the following result. For a $C$ convex and closed subset of $X$ $$\bar{x} = P_C(x) \iff \langle x-\bar{x},y-\bar{x}\rangle \le 0, \ \forall y \in C$$ So, since $M$ is a vector subspace there exist some $z \in M$ such that $z = y-\bar{x}, \ \forall y \in M$, but i dont know how to prove the equality. Any tips?

Answer 1

When $C$ is subspace $y \in C$ implies $y+\overline x \in C$ too, so we get $ \langle x-\overline x, y \rangle \leq 0$. We can now replace $y$ by $-y$ to get the reverse inequality.