Banach space theory became a recognized part of mathematical analysis with the appearance of the book of (1932). Since then the theory had a very quick development and found many significant applications, because it is primarily concerned with infinite dimensional function spaces, which arise rather naturally in applied problems. Even the nonlinear theories developed later, exploit particular structures in Banach space theory, like Hilbert spaces or more generally reflexive and/or separable Banach spaces. So any venture into the realm of modern nonlinear analysis, demands knowledge of at least the basic aspects of Banach space theory. The purpose of this chapter is to survey those parts of the theory that will equip the reader with all the necessary tools, to deal with the nonlinear problems that follow. The treatment is by no means exhaustive. Only the very basic things are presented and the knowledgeable reader will undoubtely spot important omissions. The books mentioned in the references of this chapter, which are devoted exclusively on the theory of Banach spaces, provide a more detailed treatment of the subject and the interested reader can find there more details and additional results.
KeywordsBanach Space Nonlinear Analysis Closed Subspace Topological Vector Space Weak Topology
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