Hilbert projection theorem

Remarks

The idea of the proof: let $d$ be the infimum and take $z_n \in C$ with $\norm{x - z_n} \to d$. The parallelogram law gives $\norm{z_n - z_m}^2 = 2 \norm{x - z_n}^2 + 2 \norm{x - z_m}^2 - 4 \norm{x - (z_n + z_m)/2}^2$. By convexity the midpoint lies in $C$, so the last norm is at least $d$, and $(z_n)$ is a Cauchy sequence. Completeness and closedness give a limit $y \in C$, and the same identity gives uniqueness.

When $C$ is a closed linear subspace $M$, the condition becomes $x - y \perp M$. Then $y$ is the orthogonal projection of $x$ on $M$, and $H = M \oplus M^\perp$ with the orthogonal complement $M^\perp$. For finite-dimensional $M$ this is the closest point property.

The hypotheses matter. If $C$ is not closed, such as an open ball not containing $x$, there may be no nearest point. If $C$ is not convex, such as a circle in $\R^2$ and $x$ its center, the nearest point need not be unique. In the Banach space $\R^2$ with the max norm, the nearest point of the line $\set{(t, 1)}$ to $0$ is not unique.

Going deeper

For a student

The theorem is the base of Hilbert space geometry. It gives the orthogonal decomposition $H = M \oplus M^\perp$, the existence of orthogonal projections and the Riesz representation theorem. In optimisation and approximation it shows that least squares problems over closed convex sets have unique solutions.