# Proof of Eberlein–Smulian Theorem for a reflexive Banach spaces

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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-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? – t.b. Jun 3 '12 at 9:55
@t.b.: Nothing. – Hassan Muhammad Jun 3 '12 at 9:59

## 2 Answers

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

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Thanks for the reference. – Hassan Muhammad Jun 3 '12 at 12:59

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

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