# Thread: find a closed subset of a hilbert space which has no best approximation

1. ## find a closed subset of a hilbert space which has no best approximation

Here's another one giving me no end of trouble...

Find a Hilbert space $H$ and a nonempty closed subset $K$ of $H$ such that there is $x\in H$ for which $K$ has no best approximation.
Recall that a "best approximation" of $x$ with respect to $K$ is an element $u\in K$ satisfying $\lVert x-u\rVert=\inf\{\lVert x-y\rVert:y\in K\}$.

Given what we know about best approximation, it's clear that $K$ cannot be a subspace or convex. But unfortunately I can't say much more than that at this point.

Any help will be much appreciated. Thanks !

EDIT: Nevermind, I thought of one.