Eberlein–Šmulian theorem and some of its applications

Abstract

The thesis is about Eberlein-Šmulian and some its applications. The goal is to investigate and explain different proofs of the Eberlein-Šmulian theorem. First we introduce the general theory of weak and weak* topology defined on a normed space X. Next we present the definition of a basis and a Schauder basis of a given Banach space. We give some examples and prove the main theorems which are needed to enjoy the proof of the Eberlein-Šmulian theorem given by Pelchynski in 1964. Also we present the proof given by Whitley in 1967. Next there is described the connection between the weak topology and the topology and the topology of pointwise convergence in C(K) for K compact Hausdorff. Then we give a generalization of the Eberlein-Šmulian in the context of C(K) spaces. We end the thesis by providing some examples of applications of the Eberlein-Šmulian theorem to Tauberian operators theory.