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).