Eberlein-Šmulian theorem and "Whitley's construction"

Question

The Eberlein-Šmulian theorem states that if $X$ is a Banach space, $\sigma(X,X')$ denotes the weak topology on $X$ and $A\subseteq X$, then $A$ is (relatively) $\sigma(X,X')$-compact if and only if $A$ is (relatively) sequentially $\sigma(X,X')$-compact.

Now I've read that by "Whitley's construction" (whatever that is) we can show that if $A$ is relatively $\sigma(X,X')$-compact and $x\in\overline A^{\sigma(X,\:X')}$, then there is a sequence in $A$ which converges weakly to $x$.

Honestly, I don't get that. Isn't this claim preciesely the definition of $A$ being relatively sequentially $\sigma(X,X')$-compact? Is there any subtlety I'm missing here?

Answer 1

thanks for a very good question, I hope I can shed some light on this. First of all, Whitley's construction is in Robert Whitley, An Elementary Proof of the Eberlein-Smulian Theorem, Mathematische Annalen, vol. 172, 116-118 (1967). There is free access to the article on https://gdz.sub.uni-goettingen.de

Let me follow directly Whitley's outline of 3 different notions of compactness of a Subset $A$ of a topological space $S$. He writes

A. The subset $A$ is conditionally compact if its closure is compact.

B. The subset $A$ is conditionally sequentially compact if each sequence of elements of $A$ contains a subsequence converging to an element of S, and

C. The subset $A$ is conditionally countably compact if each sequence of elements of $A$ has a cluster point in $S$.

Then Whitley explains 3 implications with strictly increasing order of sophistication, and no. 3 being the hard one with "Whitley's construction":

1. He writes "it is clear that B implies C" - conditionally sequentially compact implies conditionally countably compact - indeed, this holds for every topological space.

2. He writes "A implies B (corollary)" - conditionally compact implies conditionally sequentially compact, this is the "only if" part of your quotation. I recognize (please correct me if I am wrong) it is this part of the Theorem which you refer to as using "Whitley's contruction". However, based on the following references, I would rather call this the "Smulian part" of the Eberlein-Smulian theorem which has not a lot to do with Whitley:

It was proved in Šmulian, V., Über lineare topologische Räume, Rec. Math. [Mat. Sbornik] N. S. 7 (49), (1940). 425–448. This must be a German translation of an originally Russian publication. Unfortunately I have never found an online resource to this article, neither in German nor in Russian. If anyone should have access to this and could provide it, I will be happy to translate in English and share - I am German native and speak Russian pretty well.

Whitley argues by a variation of a standard argument about metrizability of weak topologies on separarable spaces. The argument has been given certainly hundreds of times, I share just a few places I know:

• Paul Halmos, A Hilbert space problem book, Springer Graduate Texts in Mathematics 019, (1982) 2nd edition, pages 14 with problem no. 24 and page 181 with its solution containing the definition of the critical metric. Though this is only the simple Hilbert space version of the argument, Halmos explains always very well and I recommend to take a look.

• John B. Conway, A Course in Functional Analysis, Springer Graduate Texts in Mathematics 096, (1985), pages 135-136. Conway defines the crucial metric to prove particularly that an $l^1$ sequence converging weakly converges also strongly, but the idea can be used much more broadly, a few examples being in his exercises following this chapter.

• This is also on John B. Conway, The inadequacy of sequences, The American Mathematical Monthly, Vol. 76, No. 1 (Jan., 1969), pp. 68-69, Taylor & Francis, Ltd.

• Joseph Diestel, Sequences and Series in Banach Spaces, Springer Graduate Texts in Mathematics 092, (1984), pages 17-19. says also about this part of the proof of Eberlein-Smulian "This will be accomplished in two easy steps", so it is considered the simpler part.

Whitley's outline of the "A implies B" part is certainly nice, but I doubt that it originated from him.

3. C implies A - conditionally countably compact implies conditionally compact: This is the "if" (and the hard) part of the Eberlein-Smulian Theorem as you quote it, and here comes Whitley's construction. The theorem is due to Eberlein and was originally proved in

• Eberlein, W. F., Weak compactness in Banach spaces, Proceedings of the National Academy of Sciences, vol 33 (1947), p. 51-53. I confess I did not fully understand his reasoning although I tried many times.

• Whitley, in his paper, quotes also Pelczynski, A., A proof of Eberlein-Smulian theorem by an application of basic sequences, Bull. Acad. Polon. Sci., vol 12 (1964), p. 543-548, another paper which I was unable to review. However, I recall there is a Diploma Thesis explaining this approach in detail, which I cannot find right now, please let me know if should further search and I will share.

• Whitley himself attributes the idea of his construction to Mahlon M. Day, Normed linear spaces, Springer (1958), p. 52. Unfortunately I could not review this either, but this book has an extremely well recognition, so it is certainly worth to take a look at in order to understand the essence of Whitley's construction.

In summary, Whitley's construction is used to prove the $opposite$ direction of what you quote in your question: if any sequence in $A$ converges weakly to some $x$ in the Banach space, then $A$ is relatively $\sigma(X,X')$-compact and $x\in\overline A^{\sigma(X,\:X')}$.

I conclude with quoting a paper of Hendrik Vogt arguing the compactness of the unit sphere of finite dimensional subspaces of $X$** (the $bidual$ of $X$) as used by Whitley, is actually already valid on the dual space $X$*. And then he explains Whitley's construction from this point of view. In Vogt's paper there is also a link to another original paper of Smulian.

• Hendrik Vogt, An Eberlein–Šmulian type result for the weak-star topology, Archiv der Mathematik vol. 95, pages 31–34 (2010)

I hope this helps.