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?
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?
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 $\|x\|>1, \forall x \in E.$