This single-volume textbook covers the fundamentals of linear and nonlinear functional analysis, illustrating most of the basic theorems with numerous applications to linear and nonlinear partial differential equations and to selected topics from numerical analysis and optimization theory. This book has pedagogical appeal because it features self-contained and complete proofs of most of the theorems, some of which are not always easy to locate in the literature or are difficult to reconstitute. It also offers 401 problems and 52 figures, plus historical notes and many original references that provide an idea of the genesis of the important results, and it covers most of the core topics from functional analysis. Audience: Linear and Nonlinear Functional Analysis with Applications is intended for advanced undergraduates, graduate students, and researchers and is ideal for teaching or self-study. Contents: Preface; Chapter 1: Real analysis and theory of functions: A quick review; Chapter 2: Normed vector spaces; Chapter 3: Banach spaces; Chapter 4: Inner-product spaces and Hilbert spaces; Chapter 5: The great theorems of linear functional analysis; Chapter 6: Linear partial differential equations; Chapter 7: Differential calculus in normed vector spaces; Chapter 8: Differential geometry in Rn; Chapter 9: The great theorems of nonlinear functional analysis; Bibliographical notes; Bibliography; Main notations; Index. This single-volume textbook covers the fundamentals of linear and nonlinear functional analysis, illustrating most of the basic theorems with numerous applications to linear and nonlinear partial differential equations and to selected topics from numerical analysis and optimization theory. This book has pedagogical appeal because it features self-contained and complete proofs of most of the theorems, some of which are not always easy to locate in the literature or are difficult to reconstitute. It also offers 401 problems and 52 figures, plus historical notes and many original references that provide an idea of the genesis of the important results, and it covers most of the core topics from functional analysis. **Audience:** __Linear and Nonlinear Functional Analysis with Applications__ is intended for advanced undergraduates, graduate students, and researchers and is ideal for teaching or self-study. **Contents:** Preface; Chapter 1: Real analysis and theory of functions: A quick review; Chapter 2: Normed vector spaces; Chapter 3: Banach spaces; Chapter 4: Inner-product spaces and Hilbert spaces; Chapter 5: The great theorems of linear functional analysis; Chapter 6: Linear partial differential equations; Chapter 7: Differential calculus in normed vector spaces; Chapter 8: Differential geometry in Rn; Chapter 9: The great theorems of nonlinear functional analysis; Bibliographical notes; Bibliography; Main notations; Index Functional analysis occupies a central position in modern mathematics, and has powerful applications in many other branches of the subject. This book provides a thorough and self-contained introduction to the basic aspects of both linear and nonlinear functional analysis. Most of the major results are accompanied by complete proofs, and illustrated with a variety of applications to numerical analysis, optimisation theory, and partial differential equations. As well as covering a vast amount of foundational material, historical notes and many original references are included to help the reader explore the genesis of some important results. This book is intended for advanced undergraduates, graduate students, and researchers. With illustrations and over 400 problems, it is ideal for both teaching and self-study.
