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Minimum principle is following: Let $M$ be a closed convex nonempty subset of Hilbert space. Then there exists $x\in M$ which have a minimum norm.

Assume that $M$ is not convex subset. What is a counterexample, when there have not the element with minimum norm in space $\ell^{2}$?

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    I think you mean *subset*, not subspace... In fact subspaces are always convex.2011-12-18
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    I'm guessing you still want $M$ to be closed, but will you please clarify so we don't have to guess?2011-12-18

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