Is weak topology first-countable or Lindelöf? 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?
 A: The classic paper on the Lindelöf question is by Corson (1).  I wrote a couple of papers on similar questions, too... (2,3,4)
In addition to weak topology Lindelöf, Corson investigates weak topology normal, realcompact, or paracompact.  In all cases, some Banach spaces have the property, others do not.
One of Corson's results: If $M$ is a locally compact group, then for the Banach space $C_0(M)$ (continuous functions vanishing at infinity), the following are equivalent: (a) $C_0(M)$ weak topology is normal;
(b) $C_0(M)$ weak topology is Lindelöf; (c) $M$ is metrizable.
More:  $X$ weak is paracompact iff $X$ weak is Lindelöf.
If $X^n$ weak is normal for all $n$, then $X$ weak is realcompact.


*

*Corson, H. H.,
"The weak topology of a Banach space." 
Trans. Amer. Math. Soc. 101 (1961) 1–15. 

*Edgar, G. A., "Measurability in a Banach space." Indiana Univ. Math. J. 26 (1977) 663–677. 

*Edgar, G. A., "Measurability in a Banach space. II." Indiana Univ. Math. J. 28 (1979) 559–579. 

*Edgar, G. A. & Wheeler, R. F.,
"Topological properties of Banach spaces." 
Pacific J. Math. 115 (1984) 317–350. 
