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Full Description
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?
Contents
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
