Example of weak limit not in the set

Question

Let $(E, \| \cdot \| _E)$ be a normed vector space over a field $\mathbb{K}$ and $E'$ be its dual space.

Theorem: Let $C\subset E$ be a closed (respect to the strong topology $\| \cdot \| _E$) and convex set. If a sequence $\{x_n\}\rvert_{n\in \mathbb{n\in N}}\subset C$ converges weakly to a $x\in E$, then $x\in C$.

Question: What happens if $C$ is a non convex set ? Equivalent: Find a counterexample where $C\subset E$ is a closed set (respect the strong topology $\| \cdot \| _E$) but a non convex set and
$$x_n \rightharpoonup x \notin C.$$
Is this even possible?

Note: The notation $x_n \rightharpoonup x$ means $x_n$ converges weakly to $x$.

Thanks.

Answer 1

Let $\{e_n\}$ be an orthonrmal set in Hilbert space. Then $e_n \to 0$ weakly and $\{e_1,e_2,\cdots\}$ is a closed set in the norm. The limit $0$ is not in this set.