The Best Mathematical Analysis Books for Beginners

This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas.

Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one. Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation ­ Theorem, and the gamma function are now among the book’s cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.

This book constitutes a concise introductory course on Functional Analysis for students who have studied calculus and linear algebra. The topics covered are Banach spaces, continuous linear transformations, Frechet derivative, geometry of Hilbert spaces, compact operators, and distributions. In addition, the book includes selected applications of functional analysis to differential equations, optimization, physics (classical and quantum mechanics), and numerical analysis. The book contains 197 problems, meant to reinforce the fundamental concepts.

The inclusion of detailed solutions to all the exercises makes the book ideal also for self-study. A Friendly Approach to Functional Analysis is written specifically for undergraduate students of pure mathematics and engineering, and those studying joint programmes with mathematics. Readership: Graduate students in functional analysis, operator theory and mathematical physics.

This book provides the foundations of both linear and nonlinear analysis necessary for understanding and working in twenty-first century applied and computational mathematics. In addition to the standard topics, this text includes several key concepts of modern applied mathematical analysis that should be, but are not typically, included in advanced undergraduate and beginning graduate mathematics curricula. This material is the introductory foundation upon which algorithm analysis, optimization, probability, statistics, differential equations, machine learning, and control theory are built.

When used in concert with the free supplemental lab materials, this text teaches students both the theory and the computational practice of modern mathematical analysis. Foundations of Applied Mathematics, Volume 1: Mathematical Analysis includes several key topics not usually treated in courses at this level, such as uniform contraction mappings, the continuous linear extension theorem, Daniell Lebesgue integration, resolvents, spectral resolution theory, and pseudospectra. Ideas are developed in a mathematically rigorous way and students are provided with powerful tools and beautiful ideas that yield a number of nice proofs, all of which contribute to a deep understanding of advanced analysis and linear algebra. Carefully thought out exercises and examples are built on each other to reinforce and retain concepts and ideas and to achieve greater depth. Associated lab materials are available that expose students to applications and numerical computation and reinforce the theoretical ideas taught in the text. The text and labs combine to make students technically proficient and to answer the age-old question, 'When am I going to use this?'. Audience: This textbook is appropriate for advanced undergraduate or beginning graduate students in mathematics and, potentially, graduate students in physics, engineering, statistics, or computer science. Contents: List of Notation; Foreword; Preface; Part I: Linear Analysis I; Chapter 1: Abstract Vector Spaces; Chapter 2: Linear Transformations and Matrices; Chapter 3: Inner Product Spaces; Chapter 4: Spectral Theory; Part II: Nonlinear Analysis I; Chapter 5: Metric Space Topology; Chapter 6: Differentiation; Chapter 7: Contraction Mappings and Applications; Part III: Nonlinear Analysis II; Chapter 8: Integration I; Chapter 9: Integration II; Chapter 10: Calculus on Manifolds; Chapter 11: Complex Analysis; Part IV: Linear Analysis II; Chapter 12: Spectral Calculus; Chapter 13: Iterative Methods; Chapter 14: Spectra and Pseudospectra; Chapter 15: Rings and Polynomials; Part V: Appendix; Appendix A: Foundations of Abstract Mathematics; Appendix B: The Complex Numbers and Other Fields; Appendix C: Topics in Matrix Analysis; Appendix D: The Greek Alphabet; Bibliography; Index.