How to prove the existence of a point in a compact subset of a Hilbert space with minimal inner product with a unit vector?

Question

I'm trying to solve an old qualifier problem which asks to prove the following for a compact subset $K$ of diameter $d$ in a Hilbert space $H$:

There is a point $k\in K$ and a unit vector $e$ with $<k,e>\leq<x,e>\leq<k,e>+d$ for all $x\in H$.

I know the result about the closest point of a Hilbert subspace, but I don't see how to prove this result or how to use compactness. I'd like to have some help.

Answer 1

This seems easy. Take any unit vector $e$. Consider $f: K \to \mathbb{R}$ given by $f(w) = \langle w,e \rangle$. Then since $K$ is compact and $f$ is continuous, $f$ achieves a minimum at some $k \in K$. Then for any $x \in K, \langle x,e \rangle - \langle k,e \rangle \le ||x-k|| \le d$ by Cauchy-Schwarz.