I know the following result is true in the case of strong convergence. But I don't know whether it is true in the case of weak convergence also.
Let $p>1$. Suppose that each $x\_n$ is a non negative sequence such that $\|x\_n\|\_p=1$ and $\stackrel{w}{x\_{n}\rightarrow x}$ in $\ell^p$. Is it true then that $\stackrel{w}{x\_n^p\rightarrow x^p}$ in $\ell^1$.



weak convergence
No. Consider the unit vector basis of $\ell\_2$.


