Claim:In a reflexive Banach space, the weak compactness of a subset is equivalent to the boundedness of the subset.
But there is no guarantee that the bounded subset would even have its sequences converge in the bounded subset. 1) Is this statement true how it is? 2) Is it true with the additional condition of the subset being closed?
Note: I understand that weakly compact implies bounded direction works from the definition of weak compactness having these subsequences converge in the subset.