Thread: 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 and a nonempty closed subset of such that there is for which has no best approximation.

Recall that a "best approximation" of with respect to is an element satisfying .

Given what we know about best approximation, it's clear that 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.