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Math Help - 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 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.
    Last edited by hatsoff; October 8th 2011 at 06:17 PM.
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