Charles Swartz – författare

Assuming minimal background on the part of students, this text gradually develops the principles of basic real analysis and presents the background necessary to understand applications used in such disciplines as statistics, operations research, and engineering. The text presents the first elementary exposition of the gauge integral and offers a clear and thorough introduction to real numbers, developing topics in n-dimensions, and functions of several variables. Detailed treatments of Lagrange multipliers and the Kuhn-Tucker Theorem are also presented. The text concludes with coverage of important topics in abstract analysis, including the Stone-Weierstrass Theorem and the Banach Contraction Principle.

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.

The book presents a theory of abstract duality pairs which arises by replacing the scalar field by an Abelian topological group in the theory of dual pair of vector spaces. Examples of abstract duality pairs are vector valued series, spaces of vector valued measures, spaces of vector valued integrable functions, spaces of linear operators and vector valued sequence spaces. These examples give rise to numerous applications such as abstract versions of the Orlicz–Pettis Theorem on subseries convergent series, the Uniform Boundedness Principle, the Banach–Steinhaus Theorem, the Nikodym Convergence theorems and the Vitali–Hahn–Saks Theorem from measure theory and the Hahn–Schur Theorem from summability. There are no books on the current market which cover the material in this book. Readers will find interesting functional analysis and the many applications to various topics in real analysis.

This text is an introduction to functional analysis which requires readers to have a minimal background in linear algebra and real analysis at the first-year graduate level. Prerequisite knowledge of general topology or Lebesgue integration is not required. The book explains the principles and applications of functional analysis and explores the development of the basic properties of normed linear, inner product spaces and continuous linear operators defined in these spaces. Though Lebesgue integral is not discussed, the book offers an in-depth knowledge on the numerous applications of the abstract results of functional analysis in differential and integral equations, Banach limits, harmonic analysis, summability and numerical integration. Also covered in the book are versions of the spectral theorem for compact, symmetric operators and continuous, self adjoint operators.

A functional calculus is a construction which associates with an operator or a family of operators a homomorphism from a function space into a subspace of continuous linear operators, i.e. a method for defining “functions of an operator”. Perhaps the most familiar example is based on the spectral theorem for bounded self-adjoint operators on a complex Hilbert space.This book contains an exposition of several such functional calculi. In particular, there is an exposition based on the spectral theorem for bounded, self-adjoint operators, an extension to the case of several commuting self-adjoint operators and an extension to normal operators. The Riesz operational calculus based on the Cauchy integral theorem from complex analysis is also described. Finally, an exposition of a functional calculus due to H. Weyl is given.

This text contains a basic introduction to the abstract measure theory and the Lebesgue integral. Most of the standard topics in the measure and integration theory are discussed. In addition, topics on the Hewitt-Yosida decomposition, the Nikodym and Vitali-Hahn-Saks theorems and material on finitely additive set functions not contained in standard texts are explored. There is an introductory section on functional analysis, including the three basic principles, which is used to discuss many of the classic Banach spaces of functions and their duals. There is also a chapter on Hilbert space and the Fourier transform.