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.
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Acknowledgments
I would like to thank Prof. C. W. Henson, University of Illinois at Champaign-Urbana who gave me the important reference to the work of D. Dacunha-Castelles and J. L. Krivine, and who read very carefully an earlier version of Sects. 4.1 and 4.2; Prof. E. Gordon, University of Nishni Novgorod and now Eastern Illinois University, with whom I discussed his own approach to the theory of discrete approximation; Dr. H. Ploss, University at Vienna, for many helpful discussions on the theory of strongly continuous semigroups, some of which prevented me from unsightly errors; and Prof. Dr. Eduard Emel’yanov from the Middle East Technical University at Ankara who carefully read the final version of the first edition eliminating some more misprints and errors.
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Wolff, M.P.H. (2015). Banach Spaces and Linear Operators. In: Loeb, P., Wolff, M. (eds) Nonstandard Analysis for the Working Mathematician. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7327-0_4
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