Analysis: From Concepts to Applications

This textbook covers the main results and methods of real analysis in a single volume. Taking a progressive approach to equations and transformations, this book starts with the very foundations of real analysis (set theory, order, convergence, and measure theory) before presenting powerful results that can be applied to concrete problems.

In addition to classical results of functional analysis, differential calculus and integration, Analysis discusses topics such as convex analysis, dissipative operators and semigroups which are often absent from classical treatises. Acknowledging that analysis has significantly contributed to the understanding and development of the present world, the book further elaborates on techniques which pervade modern civilization, including wavelets in information theory, the Radon transform in medical imaging and partial differential equations in various mechanical and physical phenomena.

Advanced undergraduate and graduate students, engineers as well as practitioners wishing to familiarise themselves with concepts and applications of analysis will find this book useful. With its content split into several topics of interest, the book’s style and layout make it suitable for use in several courses, while its self-contained character makes it appropriate for self-study.

Table of Contents

Chapter 1 Sets, Orders, Relations and Measures
Chapter 2 Encounters With Limits
Chapter 3 Elements of Functional Analysis
Chapter 4 Hilbert Spaces
Chapter 5 The Power of Differential Calculus
Chapter 6 A Touch of Convex Analysis
Chapter 7 Integration
Chapter 8 Differentiation and Integration
Chapter 9 Partial Differential Equations
Chapter 10 Evolution Problems
Appendix: The Brouwer’s Fixed Point Theorem