The sequence is weakly convergent in but it's not strongly convergent in .
thank you for your hints. By definition, a sequence in a banach space is weakly convergent to an element , if for every , where denotes the dual space of .
The dual space of is the space of all linear functions . So I have to prove that there exists a , with for all ?
How to find this ?
Now to the other statement:
Do I have to prove that there is no with ?