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



Weak versus strong convergence
No, of course not. Take two different sequences converging weakly to zero and interleave them.


