Applied Algebra and Functional Analysis: (Dover Books on Mathematics)

""The treatment is thorough; the book will certainly serve as a valuable reference."" - American Scientist. Written for graduate and advanced undergraduate students in engineering and science, this book focuses primarily on set theory, algebra, and analysis. Useful both for self-study and as a reference, the text provides the readers with appropriate mathematical background for advanced study in engineering or science, a basis for more advanced mathematical study, and an overview of modern mathematics.The book is essentially divided into three parts: set theory (Chapter 1), algebra (Chapters 2‒4), and analysis (Chapters 5‒7). The first two chapters deal with the fundamental concepts of sets, functions, relations, and equivalence relations, and algebraic structures. Chapters 3 and 4 cover vector spaces and linear transformations, and finite-dimensional vector spaces and matrices. The last three chapters investigate metric spaces, normed spaces (Banach spaces) and inner product spaces (Hilbert spaces), and linear operators. A generous number of exercises have been integrated into the text and a section of references and notes is found at the end of each chapter. 1981 edition.

Table of Contents:
PREFACE CHAPTER 1: FUNDAMENTAL CONCEPTS 1.1 Sets 1.2 Functions 1.3 Relations and Equivalence Relations 1.4 Operations on Sets 1.5 Mathematical Systems Considered in This Book 1.6 References and Notes References CHAPTER 2: ALGEBRAIC STRUCTURES 2.1 Some Basic Structures of Algebra A. Semigroups and Groups B. Rings and Fields C. "Modules, Vector Spaces, and Algebras" D. Overview 2.2 Homomorphisms 2.3 Application to Polynomials 2.4 References and Notes References CHAPTER 3: VECTOR SPACES AND LINEAR TRANSFORMATIONS 3.1 Linear Spaces 3.2 Linear Subspaces and Direct Sums 3.3 "Linear Independence, Bases, and Dimension" 3.4 Linear Transformations 3.5 Linear Functionals 3.6 Bilinear Functionals 3.7 Projections 3.8 Notes and References References CHAPTER 4: FINITE-DIMENSIONAL VECTOR SPACES AND MATRICES 4.1 Coordinate Representation of Vectors 4.2 Matrices A. Representation of Linear Transformations by Matrices B. Rank of a Matrix C. Properties of Matrices 4.3 Equivalence and Similarity 4.4 Determinants of Matrices 4.5 Eigenvalues and Eigenvectors 4.6 Some Canonical Forms of Matrices 4.7 "Minimal Polynomials, Nilpotent Operators and the Jordan Canonical Form" A. Minimal Polynomials B. Nilpotent Operators C. The Jordan Canonical Form 4.8 Bilinear Functionals and Congruence 4.9 Euclidean Vector Spaces A. Euclidean Spaces : Definition and Properties B. Orthogonal Bases 4.10 Linear Transformations on Euclidean Vector Spaces A. Orthogonal Transformations B. Adjoint Transformations C. Self-Adjoint Transformations D. Some Examples E. Further Properties of Orthogonal Transformations 4.11 Applications to Ordinary Differential Equations A. Initial-Value Problem : Definition B. Initial-Value Problem : Linear Systems 4.12 Notes and References References CHAPTER 5: METRIC SPACES 5.1 Definition of Metric Spaces 5.2 Some Inequalities 5.3 Examples of Important Metric Spaces 5.4 Open and Closed Sets 5.5 Complete Metric Spaces 5.6 Compactness 5.7 Continuous Functions 5.8 Some Important Results in Applications 5.9 Equivalent and Homeomorphic Metric Spaces. Topological Spaces 5.10 Applications A. Applications of the Contraction Mapping Principle B. Further Applications to Ordinary Differential Equations 5.11 References and Notes References CHAPTER 6: NORMED SPACES AND INNER PRODUCT SPACES 6.1 Normed Linear Spaces 6.2 Linear Subspaces 6.3 Infinite Series 6.4 Convex Sets 6.5 Linear Functionals 6.6 Finte-Dimensional Spaces 6.7 Geometric Aspects of Linear Functionals 6.8 Extension of Linear Functionals 6.9 Dual Space and Second Dual Space 6.10 Weak Convergence 6.11 Inner Product Spaces 6.12 Orthogonal Complements 6.13 Fourier Series 6.14 The Riesz Representation Theorem 6.15 Some Applications A. Approximation of Elements in Hilbert Space (Normal Equations) B. Random Variables C. Estimation of Random Variables 6.16 Notes and References References CHAPTER 7: LINEAR OPERATORS 7.1 Bounded Linear Transformations 7.2 Inverses 7.3 Conjugate and Adjoint Operators 7.4 Hermitian Operators 7.5 "Other Linear Operators: Normal Operators, Projections, Unitary Operators, and Isometric Operators" 7.6 The Spectrum of an Operator 7.7 Completely Continuous Operators 7.8 The Spectral Theorem for Completely Continuous Normal Operators 7.9 Differentiation of Operators 7.10 Some Applications A. Applications to Integral Equations B. An Example from Optimal Control C. Minimization of Functionals: Method of Steepest Descent 7.11 References and Notes References Index