Banach Spaces and Linear Operators

Abstract

In this chapter we deal with old and new applications of nonstandard analysis to the theory of Banach spaces and linear operators. In particular we consider the structure theory of Banach spaces, basic operator theory, strongly continuous semigroups of operators, approximation theory of operators and their spectra, and the Fixed Point Property. To include in this chapter interesting examples of nonstandard functional analysis we must assume that the reader is familiar with the basics of Banach spaces and operator theory. Non-experts in these field can, however, profit from this chapter by looking at the elementary applications with which we begin every section. Moreover we refer to the book (Siu-Ah Ng, JAMA Nonstandard methods in functional analysis, 2010, [45]) for those, who want to learn functional analysis and simultaneously nonstandard analysis.