We will give a brief proof of the Eberlein-Smulian theorem
which says that for a subset of a Banach space endowed with the weak
topology three conditions are equivalent, namely: conditional
compactness, conditional sequential compactness and conditional
countable compactness. The proof will use standard results from theory
of Banach spaces, namely Alaoglu's theorem and the Hahn-Banach theorem.
The talk is based on the article *An Elementary Proof of
Eberlein-Smulian Theorem* by Robert Whitley (Math. Ann. 1967).