Looking for the proof of Eberlein-Smulian Theorem.

Searching for the proof is what I break with this morning. Some of my friends recommend Haim Brezis (Functional Analysis, Sobolev Spaces and Partial Differential Equations). After I search the book, I only found the statement of the theorem, is the proof very difficult to grasp? Why is Haim Brezis skip it in his book?

Please I need a reference where I can find the proof in detail.

Theorem:(Eberlein-Smul'yan Theorem) A Banach space $E$ is reflexive if and only if every (norm) bounded sequence in $E$ has a subsequence which converges weakly to an element of $E$.

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    $\begingroup$ -1: I quote from Brézis right after stating the theorem as Theorem 3.19: The proof of Theorem 3.19 is rather delicate and is omitted; see e.g. R. Holmes [1], K. Yosida [1], N. Dunford-J.T. Schwartz [1], Diestel [2], or Problem 10. What more do you want? $\endgroup$ – t.b. Jun 3 '12 at 9:55
  • 1
    $\begingroup$ @t.b.: Nothing. $\endgroup$ – Hassan Muhammad Jun 3 '12 at 9:59

I made this answer CW, so that other people can add further references if they think it's suitable.


Kôsaku Yosida, Functional Analysis, Springer 1980, Chapter V, Appendix, section 4. (This appears to be the 6th edition).


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