Let $X$ be a reflexive Banach space and $K$ a nonempty closed convex subset of $X$. prove that there exists an $x\in K$ such that $\|x\|=\inf\limits_{y\in K}\|y\|$.
I try to prove it in the way that $X$ is a Hilbert space but I fail because Parallelogram law for Hilbert space is not true here.
There is a sequence $\{x_n\} \subset K$ such that $\|x_n\| \to m$ where $m=\inf \{\|x\|: x\in K\}$. By Eberlein Smulian Theorem there is a subsequence $x_{n_k}$ converging weakly to some point $x$. Since convex closed sets are weakly closed we get $ x \in K$. Now $\|x\| \leq \lim \inf \|x_{n_k}\| =m\leq \|x\|$ which implies $\|x\|=m$.
[ The following property has been used: $u_n \to u$ weakly implies $\|u\| \leq \lim \inf \|u_n\|$. This is easy because for every continuous linear functional $f$ of norm $1$ we have $|f(u)| =\lim |f(u_n)| \leq \lim \inf \|u_n\|$].
