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Let $X$ be a normed space. I want to prove that for any proper linear subspace $M$, there exists a point $x$ with $||x||=1$ such that $inf\{||x-y||:y\in M\}>1-\epsilon$ for arbitrary epsilon. Could anyone show me how to do this?

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Do you mean $>1-\epsilon$? Otherwise you can just take $x\in M$. Or maybe you specifically want to find an $x$ not lying in $M$? – froggie Nov 12 '12 at 20:35
Sorry, that was a typo. I meant greater than not less than. – Parakee Nov 12 '12 at 20:36
up vote 3 down vote accepted

This is well known Riesz lemma. For the proof see this notes.

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