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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?

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This sequence has convergent subsequences, but their limits depend on the chosen subsequence. Here is 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 $0$.

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