Does any closed subset in a Banach space have (at least one) point that has a minimum norm?
I think this statement is obviously true, but how do I prove its correctness?
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The previous version of my answer was wrong. The following example was suggested by commenter(see the comments).
The answer to this question is no. Consider the subset $$E=\left\{\left(1+\frac{1}{n}\right)e_n:n \in \mathbb{N}\right\} \subseteq \ell_2$$
where $e_n=(0,0,\ldots,0,1,0,\ldots) \in \ell_2$.
$E$ is closed subset of $\ell_2, \ \inf\left\{\|x\|:x\in E\right\}=1$ and $\|x\|>1, \forall x \in E.$
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Try to apply your hint to the closed subset $$ \left{(1+1/n)e_n : n \in \mathbb{N}\right} \subseteq \ell_2 $$ where $(e_n)$ is the standard orthonormal basis. – commenter Nov 28 '12 at 09:19
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Thanks for fixing your answer. By the way: a refinement of your original hint does work if you assume $E$ to be closed and convex and the Banach space is uniformly convex, for example $L^p(\mu)$ spaces for $1 \lt \mu \lt \infty$. Maybe the Hilbert space version of this result is what you had in mind. – commenter Nov 28 '12 at 14:28