Linear Functional Analysis for Scientists and Engineers

"This book provides a concise and meticulous introduction to functional analysis. Since the topic draws heavily on the interplay between the algebraic structure of a linear space and the distance structure of a metric space, functional analysis is increasingly gaining the attention of not only mathematicians but also scientists and engineers. The purpose of the text is to present the basic aspects of functional analysis to this varied audience, keeping in mind the considerations of applicability. A novelty of this book is the inclusion of a result by Zabreiko, which states that every countably subadditive seminorm on a Banach space is continuous. Several major theorems in functional analysis are easy consequences of this result. The entire book can be used as a textbook for an introductory course in functional analysis without having to make any specific selection from the topics presented here. Basic notions in the setting of a metric space are defined in terms of sequences. These include total boundedness, compactness, continuity and uniform continuity. Offering concise and to-the-point treatment of each topic in the framework of a normed space and of an inner product space, the book represents a valuable resource for advanced undergraduate students in mathematics, and will also appeal to graduate students and faculty in the natural sciences and engineering. The book is accessible to anyone who is familiar with linear algebra and real analysis."--Publisher website Preface 5 Contents 7 About the Author 9 Symbols and Abbreviations 11 1 Prerequisites 15 1.1 Relations on a Set 15 1.2 Linear Spaces and Linear Maps 17 1.3 Metric Spaces and Continuous Functions 21 1.4 Lebesgue Measure and Integration 33 2 Basic Framework 47 2.1 Normed Spaces 47 2.2 Inner Product Spaces 58 2.3 Banach Spaces 67 2.4 Hilbert Spaces 76 3 Bounded Linear Maps 89 3.1 Continuity of a Linear Map 89 3.2 Zabreiko Theorem and Uniform Boundedness 103 3.3 Closed Graph and Open Mapping Theorems 110 3.4 Compact Linear Maps 120 4 Dual Spaces, Transposes and Adjoints 133 4.1 Hahn--Banach Theorems 133 4.2 Dual Spaces and Their Representations 142 4.3 Transposes and Adjoints 150 4.4 Self-Adjoint, Normal and Unitary Operators 160 5 Spectral Theory 173 5.1 Spectrum of a Bounded Operator 173 5.2 Spectrum of a Compact Operator 185 5.3 Spectra of Transposes and Adjoints 192 5.4 Spectral Theorem 199 Solutions to Exercises 217 References 263 Index 265 Front Matter....Pages i-xiv Prerequisites....Pages 1-32 Basic Framework....Pages 33-74 Bounded Linear Maps....Pages 75-118 Dual Spaces, Transposes and Adjoints....Pages 119-158 Spectral Theory....Pages 159-202 Back Matter....Pages 203-254