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# Weak verses strong convergence

This is my first time posting.

I am well aware that an $L^2$ weakly converging sequence is not convergent in the corresponding strong topology. However, my question is as follows, do the sequence of norms corresponding to a weakly convergent sequence converge?

Take for instance the sine function on (0,1), specifically $\sin(x/\varepsilon)$, this weakly converges to zero, and the norms converge to the mean of $|\sin^2|$.

So despite no strong convergence, do the norms still converge to something else?

Many thanks for you help and time in advance,

Daniel