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An introduction to functional analysis
Swartz, Charles, 1938-New York : M. Dekker, c1992.Based on an introductory, graduate-level course given by Swartz at New Mexico State University, this textbook, written for students with a moderate knowledge of point set topology and integration theory, explains the principles and theories of functional analysis and their applications.
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Introduction to gauge integrals
Swartz, Charles, 1938-Singapore ; River Edge, NJ : World Scientific, ©2001.This book presents the Henstock/Kurzweil integral and the McShane integral. These two integrals are obtained by changing slightly the definition of the Riemann integral. These variations lead to integrals which are much more powerful than the Riemann integral. The Henstock/Kurzweil integral is an unconditional integral for which the fundamental theorem of calculus holds in full generality, while the McShane integral is equivalent to the Lebesgue integral in Euclidean spaces. A basic knowledge of introductory real analysis is required of the reader, who should be familiar with the fundamental properties of the real numbers, convergence, series, differentiation, continuity, etc.This book presents the Henstock/Kurzweil integral and the McShane integral. These two integrals are obtained by changing slightly the definition of the Riemann integral. These variations lead to integrals which are much more powerful than the Riemann integral. The Henstock/Kurzweil integral is an unconditional integral for which the fundamental theorem of calculus holds in full generality, while the McShane integral is equivalent to the Lebesgue integral in Euclidean spaces.A basic knowledge of introductory real analysis is required of the reader, who should be familiar with the fundamental properties of the real numbers, convergence, series, differentiation, continuity, etc.
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Multiplier convergent series
Swartz, Charles, 1938-Hackensack, N.J. : World Scientific Pub., ©2009.If [symbol] is a space of scalar-valued sequences, then a series [symbol] xj in a topological vector space X is [symbol]-multiplier convergent if the series [symbol] tjxj converges in X for every [symbol]. This monograph studies properties of such series and gives applications to topics in locally convex spaces and vector-valued measures. A number of versions of the Orlicz-Pettis theorem are derived for multiplier convergent series with respect to various locally convex topologies. Variants of the classical Hahn-Schur theorem on the equivalence of weak and norm convergent series in [symbol] are also developed for multiplier convergent series. Finally, the notion of multiplier convergent series is extended to operator-valued series and vector-valued multipliers.If is a space of scalar-valued sequences, then a series j xj in a topological vector space X is -multiplier convergent if the series j=1 tjxj converges in X for every {tj} . This monograph studies properties of such series and gives applications to topics in locally convex spaces and vector-valued measures. A number of versions of the Orlicz-Pettis theorem are derived for multiplier convergent series with respect to various locally convex topologies. Variants of the classical Hahn-Schur theorem on the equivalence of weak and norm convergent series in 1 are also developed for multiplier convergent series. Finally, the notion of multiplier convergent series is extended to operator-valued series and vector-valued multipliers.
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