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 $E$.
Best Answer
I think Megginson's book An Introduction to Banach Space Theory (GTM 183) is worth having look at. You can preview some parts of the book at Google Books. (I believe the whole section 2.8 Weak Compactness might be interesting for you.)
Albiac–Kalton, Topics in Banach Space Theory (GTM 233), Corollary 1.6.4, page 24.
Whitley, An elementary proof of the Eberlein–Šmulian theorem, Mathematische Annalen 172 (2), 1967, 116–118. Freely available from GDZ.
Zeidler, Nonlinear functional analysis and its applications. II/A: Linear monotone operators, Springer, 1990, Theorem 21.D
I made this answer CW, so that other people can add further references if they think it's suitable.