Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, let $\mathbb{N}$ be the set of natural numbers, and let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$ and zero vector $0_{H}$. For a subset $A\subseteq H$ let $A^{\perp}$ be its orthogonal complement. Every closed linear subspace $N$ of $H$ is nonempty and convex, and is closed by hypothesis, so that Nearest-Point Projection onto a Nonempty Closed Convex Subset of a Real Hilbert Space applies to $K=N$; we write $P_{N}:H\to H$ for the map $x\mapsto P_{N}x$ of Nearest-Point Projection onto a Nonempty Closed Convex Subset of a Real Hilbert Space §existence. Let $M$ be a closed linear subspace of $H$. Then the following hold (and, since $M$ is arbitrary, they hold for every closed linear subspace in place of $M$).

1. (Characterisation)¶ For $x\in H$ and $z\in M$, $z=P_{M}x$ if and only if $x-z\in M^{\perp}$.

2. (Linearity and idempotence)¶ $P_{M}$ is a linear map, $P_{M}x=x$ for every $x\in M$, and $P_{M}(P_{M}x)=P_{M}x$ for every $x\in H$.

3. (Pythagoras and contraction)¶ For every $x\in H$, $|x|^{2}=|P_{M}x|^{2}+|x-P_{M}x|^{2}$; consequently $|P_{M}x|\le|x|$ and $|x-P_{M}x|\le|x|$.

4. (The complement)¶ $M^{\perp}$ is a closed linear subspace of $H$, $M\cap M^{\perp}=\{0_{H}\}$, $x-P_{M}x=P_{M^{\perp}}x$ for every $x\in H$, and $(M^{\perp})^{\perp}=M$.

5. (Decomposition)¶ Every $x\in H$ can be written in exactly one way as $x=m+m'$ with $m\in M$ and $m'\in M^{\perp}$, namely with $m=P_{M}x$ and $m'=x-P_{M}x$.

6. (Density criterion)¶ A linear subspace $L$ of $H$ is dense in $H$ if and only if $L^{\perp}=\{0_{H}\}$.