Unique Point of Minimal Distance to Closed Convex Subset of Hilbert Space

Theorem
Let $H$ be a Hilbert space, and let $h \in H$.

Let $K \subseteq H$ be a closed, convex, non-empty subset of $H$.

Then there is a unique point $k_0 \in K$ such that:


 * $\left\|{h - k_0}\right\| = d \left({h, K}\right)$

where $d$ denotes distance to a set.

Furthermore, if $K$ is a linear subspace, this point is characterised by:


 * $\left\|{h - k_0}\right\| = d \left({h, K}\right) \iff \left({h - k_0}\right) \perp K$

where $\perp$ signifies orthogonality.