Understanding the relation of weak and weak star toplogy

Question

I'm working with Eberlein- Smulian Theorem fromm the book "Topics in Banach Space Theory". During the proof I have seen that there is used a lot the concept of weak topology and weak star topology. For a given Banach space X according to the definitions the weak topology is defined on X and the weak star topology is defined on the dual space X* . While trying to understand the proof I see that for a set A in X there is considered the weak* closure W of A in X** . *I don't understand how we can speak about the weak and weak-star closure of a set A in X, since the weak-star closure, to me, makes sense only for sets in X **. I am really struggling with this issue! Is anybody who can tell me which is the clue here? The set A to which I am referring is used to prove that (iii) implies (i) in the last paragraph of the theorem. Any advice would be really appreciated!

Answer 1

The short answer relies on two facts:

1. $X^{**}$ is a dual space, so it has a weak* topology.
2. $X$ can be considered a subspace of $X^{**}$.

As a side remark, the weak topology on $X$ coincides with the subscpace topology inherited from $X^{**}$ with the weak* topology, since both are derived from the functionals in $X^*$.