I'm assuming that this is meant to take place in the context of a Banach space.
For the counterexample, take to be the space of sequences converging to zero, with the sup norm. (The dual space can be identified with , the space of all sequences with absolutely convergent sum.) Let be the sequence with a 1 in the n'th coordinate and zeros everywhere else. Then weakly but not strongly.
However, a similar example shows that the result claimed in the original problem is false. In fact, take to be the sequence with a 1 in the first and n'th coordinates and zeros everywhere else: if k=1 or n, and for all other values of k. Then weakly but not strongly, and .
Edit. The result is true in the case where is a Hilbert space. Is that what is intended here?