Foundations of Applied Mathematics, Volume 1: Mathematical Analysis

✍ Scribed by Jeffrey Humpherys, Tyler J. Jarvis, Emily J. Evans

• Publisher: SIAM-Society for Industrial and Applied Mathematics
• Year: 2017
• Tongue: English
• Leaves: 707
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents
List of Notation
Preface
Part I. Linear Analysis I
1. Abstract Vector Spaces
1.1 Vector Algebra
1.2 Spans and Linear Independence
1.3 Products, Sums, and Complements
1.4 Dimension, Replacement, and Extension
1.5 Quotient Spaces
Exercises
2. Linear Transformations and Matrices
2.1 Basics of Linear Transformations I
2.2 Basics of Linear Transformations II
2.3 Rank, Nullity, and the First Isomorphism Theorem
2.4 Matrix Representations
2.5 Composition, Change of Basis, and Similarity
2.6 Important Example: Bernstein Polynomials
2.7 Linear Systems
2.8 Determinants I
2.9 Determinants II
Exercises
3. Inner Product Spaces
3.1 Introduction to Inner Products
3.2 Orthonormal Sets and Orthogonal Projections
3.3 Gram-Schmidt Orthonormalization
3.4 QR with Householder Transformations
3.5 Normed Linear Spaces
3.6 Important Norm Inequalities
3.7 Adjoints
3.8 Fundamental Subspaces of a Linear Transformation
3.9 Least Squares
Exercises
4. Spectral Theory
4.1 Eigenvalues and Eigenvectors
4.2 Invariant Subspaces
4.3 Diagonalization
4.4 Schur's Lemma
4.5 The Singular Value Decomposition
4.6 Consequences of the SVD
Exercises
Part II. Nonlinear Analysis I
5. Metric Space Topology
5.1 Metric Spaces and Continuous Functions
5.2 Continuous Functions and Limits
5.3 Closed Sets, Sequences, and Convergence
5.4 Completeness and Uniform Continuity
5.5 Compactness
5.6 Uniform Convergence and Banach Spaces
5.7 The Continuous Linear Extension Theorem
5.8 Topologically Equivalent Metrics
5.9 Topological Properthes
5.10 Banach-Valued Integration
Exercises
6. Differentiation
6.1 The Directional Derivative
6.2 The Fréchet Derivative in Rn
6.3 The General Fréchet Derivative
6.4 Properties of Derivatives
6.5 Mean Value Theorem and Fundamental Theorem of Calculus
6.6 Taylor's Theorem
Exercises
7. Contraction Mappings and Applications
7.1 Contraction Mapping Principle
7.2 Uniform Contraction Mapping Principle
7.3 Newton's Method
7.4 The Implicit and Inverse Function Theorems
7.5 Conditioning
Exercises
Part Ill. Nonlinear Analysis II
8. Integration I
8.1 Multivariable Integration
8.2 Overview of Daniell - Lebesgue Integration
8.3 Measure Zero and Measurability
8.4 Monotone Convergence and Integration on Unbounded Domains
8.5 Fatou's Lemma and the Dominated Convergence Theorem
8.6 Fubini's Theorem and Leibniz's Integral Rule
8.7 Change of Variables
Exercises
9. Integration II
9.1 Every Normed Space Has a Unique Completion
9.2 More about Measure Zero
9.3 Lebesgue-Integrable Functions
9.4 Proof of Fubini's Theorem
9.5 Proof of the Change of Variables Theorem
Exercises
10. Calculus on Manifolds
10.1 Curves and Arclength
10.2 Line Integrals
10.3 Parametrized Manifolds
10.4 Integration on Manifolds
10.5 Green's Theorem
Exercises
11. Complex Analysis
11.1 Holomorphic Functions
11.2 Properties and Examples
11.3 Contour Integrals
11.4 Cauchy's Integral Formula
11.5 Consequences of Cauchy's Integral Formula
11.6 Power Series and Laurent Series
11.7 The Residue Theorem
11.8 The Argument Principle and Its Consequences
Exercises
Part IV. Linear Analysis II
12. Spectral Calculus
12.1 Projections
12.2 Generalized Eigenvectors
12.3 The Resolvent
12.4 Spectral Resolution
12.5 Spectral Decomposition I
12.6 Spectral Decomposition II
12.7 Spectral Mapping Theorem
12.8 The Perron-Frobeniius Theorem
12.9 The Drazin Inverse
12.10 Jordan Canonical Form
Exercises
13. Iterative Methods
13.1 Methods for Linear Systems
13.2 Minimal Polynomials and Krylov Subspaces
13.3 The Arnoldi Iteration and GMRES Methods
13.4 Computing Eigenvalues I
13.5 Computing Eigenvalues II
Exercises
14. Spectra and Pseudospectra
14.1 The Pseudospectrum
14.2 Asymptotic and Transient Behavior
14.3 *Proof of the Kreiss Matrix Theorem
Exercises
15. Rings and Polynomials
15.1 Definition and Examples
15.2 Euclidean Domains
15.3 The Fundamental Theorem of Arithmetic
15.4 Homomorphisms
15.5 Quotients and the First Isomorphism Theorem
15.6 The Chinese Remainder Theorem
15.7 Polynomial Interpolation and Spectral Decomposition
Exercises
Part V. Appendices
A. Foundations of Abstract Mathematics
A.1 Sets and Relations
A.2 Functions
A.3 Orderings
A.4 Zorn's Lemma, the Axiom of Choice, and Well Ordering
A.5 Cardinality
B. The Complex Numbers and Other Fields
B.1 Complex Numbers
B.2 Fields
C. Topics in Matrix Analysis
C.1 Matrix Algebra
C.2 Block Matrices
C.3 Cross Products
D. The Greek Alphabet
Bibliography
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Index
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