# Question about weak convergence in Banach space

Let $$X$$ be Banach space and $$(x_n)_n\subset X$$ a weakly convergent sequence. My question is if one can conlude then that the sequence $$y_n:=||x_n||x_n$$ is also weakly convergent or if there exists a counterexample?

Let $$X=l^2$$, $$x_{2n}=e_1$$, $$x_{2n+1} = e_1 + e_{2n+1}$$. Then $$x_n\rightharpoonup e_1$$, $$y_{2n} \rightharpoonup e_1$$, $$y_{2n+1} \rightharpoonup \sqrt2 e_1$$.
If $$x_n \rightharpoonup 0$$ then $$y_n \rightharpoonup 0$$: for $$f\in X^*$$ $$f(y_n)= \|x_n\|f(x_n)\to0$$ because weakly convergent sequences are bounded. This makes it hard to find a counterexample because all canonical examples of weakly and not strongly converging sequences have limit $$0$$.