find a closed subset of a hilbert space which has no best approximation
Here's another one giving me no end of trouble...
Recall that a "best approximation" of with respect to is an element satisfying .
Find a Hilbert space
and a nonempty closed subset
such that there is
has no best approximation.
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.