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?

  • $\begingroup$ It's certainly not first countable for infinite dimensional spaces. Your statement is correct, it's the Eberlein-Smulian theorem (you should say "relatively weakly compact", though). $\endgroup$ – David Mitra Jan 9 '15 at 16:05
  • $\begingroup$ See this for the assertion I made in the previous comment. $\endgroup$ – David Mitra Jan 9 '15 at 16:11
  • $\begingroup$ awesome! Thanks! But, forgive my ignorance, you said "It's certainly not first countable for infinite dimensional spaces", how to prove it or which book I could find it in? Thanks again! $\endgroup$ – Eric Yau Jan 9 '15 at 16:23
  • $\begingroup$ See the answer in the link in my previous comment. $\endgroup$ – David Mitra Jan 9 '15 at 16:24

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.

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

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

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

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

  • $\begingroup$ Could you provide me the first article ("The weak topology of a Banach space.")? $\endgroup$ – Eric Yau Jan 9 '15 at 17:26
  • $\begingroup$ Transactions (and other AMS journals) free online after 5 years ... ams.org/jourcgi/jrnl_toolbar_nav/tran_all $\endgroup$ – GEdgar Jan 9 '15 at 18:26
  • $\begingroup$ Thanks! Just asking, are you Edgar, G. A. himself? $\endgroup$ – Eric Yau Jan 9 '15 at 18:55

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