Let X be a Banach space and $X^*$ is separable. Show that if K is a weakly compact subset of X, then K with the relative weak topology is metrizable.
I can easily show that K with the weak topology is metrizable. Is it sufficient?
I think about a homeomorphism between K and the closed Ball X, then closed Ball X is weakly compact, which show that X is reflexive. Now by reflexivity of $X^*$, Ball X and K with the relative weak topology are metrizable. but I can not define a homeomorphism. Also I think K can be finite and in this case there isn't any homeomorhism.
Please help me. Thanks so much