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

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

    Given what we know about best approximation, it's clear that $\displaystyle K$ 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.
    Last edited by hatsoff; Oct 8th 2011 at 05:17 PM.
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