Is the weak topology on a Banach space( or normed linear space ) first countable or Lindelöf? If not, which condition should be added?
Definition: A topology space is said to be Lindelöf if its any open-cover has a countable subcover.
"A subset, say, X, of a Banach space( or normed linear space ) is weakly compact iff any sequence of X has a subsequence converging weakly to some element in X", is this statement right?
We need first countability and Lindelöf condition for the equivalence between compactness and sequential compactness, don't we?