Nearest-Point Projection onto a Nonempty Closed Convex Subset 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 $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$, let $K\subseteq H$ be nonempty, convex and closed. Then the following hold.

1. (Existence and uniqueness of a nearest point)(i) For every $x\in H$ there is exactly one point of $K$, denoted $P_{K}x$, such that $|x-P_{K}x|\le|x-y|$ for every $y\in K$.

2. (Variational characterisation)(ii) For $x\in H$, a point $z\in K$ satisfies $z=P_{K}x$ if and only if $\langle x-z,\,y-z\rangle\le 0$ for every $y\in K$.

3. (Fixed points)(iii) For $x\in H$, $P_{K}x=x$ if and only if $x\in K$.

4. (Nonexpansiveness)(iv) For all $x,x'\in H$, $|P_{K}x-P_{K}x'|\le|x-x'|$.