# nonempty closed convex subset of reflexive Banach space achieves its minimum norm

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 $$|f(u)| =\lim |f(u_n)| \leq \lim \inf \|u_n\|$$].