Analysis in vector spaces : a course in advanced calculus

The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences.

The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory of manifolds. Additional topical coverage includes:

Sets and functions Real numbers Vector functions Normed vector spaces First- and higher-order derivatives Diffeomorphisms and manifolds Multiple integrals Integration on manifolds Stokes' theorem Basic point set topology

Numerous examples and exercises are provided in each chapter to reinforce new concepts and to illustrate how results can be applied to additional problems. Furthermore, proofs and examples are presented in a clear style that emphasizes the underlying intuitive ideas. Counterexamples are provided throughout the book to warn against possible mistakes, and extensive appendices outline the construction of real numbers, include a fundamental result about dimension, and present general results about determinants.

Assuming only a fundamental understanding of linear algebra and single variable calculus, Analysis in Vector Spaces is an excellent book for a second course in analysis for mathematics, physics, computer science, and engineering majors at the undergraduate and graduate levels. It also serves as a valuable reference for further study in any discipline that requires a firm understanding of mathematical techniques and concepts.

Author Notes

Mustafa A. Akcoglu, PhD, is Professor Emeritus in the Department of Mathematics at the University of Toronto, Canada. He has authored or coauthored over sixty journal articles on the topics of ergodic theory, functional analysis, and harmonic analysis.
Paul F.A. Bartha, PhD, is Associate Professor in the Department of Philosophy at. The University of British Columbia, Canada. He has authored or coauthored journal articles on topics such as probability and symmetry, probabilistic paradoxes, and the general philosophy of science.
Dzung Minh Ha, PHD, is Associate Professor in the Department of Mathematics at Ryerson University, Canada. Dr. Ha focuses his research in the areas of ergodic theory operator theory.

Reviews 1

Choice Review

This work by Akcoglu (emer., Univ. of Toronto), Bartha (Univ. of British Columbia), and Ha (Ryerson Univ.) combines features of introductory real analysis books (e.g., basics of sets, functions, structure and topology of the real line), with some material on linear algebra and normed vector spaces. After those basics, the book addresses calculus on vector spaces, but at a level well beyond the multivariate calculus appearing in standard "fat" calculus texts such as James Stewart's Calculus (6th ed., 2008) in order to provide the third course in a calculus sequence. The book discusses derivatives of vector-valued functions, and also includes chapters such as "Diffeomorphisms and Manifolds," "Multiple Integrals" (introduced through Jordan sets and volume), "Integration on Manifolds," and "Stokes' Theorem." The authors do not shy away from doing the hard work involved in proving say, the change of variable theorem for integration, the inverse function theorem, and Stokes's theorem--work which is not generally seen in standard calculus books--and thus they are quite correct when they state that students need a firm grip on single-variable calculus and some linear algebra, and a good comfort level with the comprehension and construction of rigorous proofs. Includes many examples and an excellent selection of exercises. Summing Up: Recommended. Upper-division undergraduates and graduate students. D. Robbins Trinity College (CT)

Preface	p. ix
Part I Background Material
1 Sets and Functions	p. 3
1.1 Sets in General	p. 3
1.2 Sets of Numbers	p. 10
1.3 Functions	p. 17
2 Real Numbers	p. 31
2.1 Review of the Order Relations	p. 32
2.2 Completeness of Real Numbers	p. 36
2.3 Sequences of Real Numbers	p. 40
2.4 Subsequences	p. 45
2.5 Series of Real Numbers	p. 50
2.6 Intervals and Connected Sets	p. 54
3 Vector Functions	p. 61
3.1 Vector Spaces: The Basics	p. 62
3.2 Bilinear Functions	p. 82
3.3 Multilinear Functions	p. 88
3.4 Inner Products	p. 95
3.5 Orthogonal Projections	p. 103
3.6 Spectral Theorem	p. 109
Part II Differentiation
4 Normed Vector Spaces	p. 123
4.1 Preliminaries	p. 124
4.2 Convergence in Normed Spaces	p. 128
4.3 Norms of Linear and Multilinear Transformations	p. 135
4.4 Continuity in Normed Spaces	p. 142
4.5 Topology of Normed Spaces	p. 156
5 Derivatives	p. 175
5.1 Functions of a Real Variable	p. 176
5.2 Differentiable Functions	p. 190
5.3 Existence of Derivatives	p. 201
5.4 Partial Derivatives	p. 205
5.5 Rules of Differentiation	p. 211
5.6 Differentiation of Products	p. 218
6 Diffeomorphisms and Manifolds	p. 225
6.1 The Inverse Function Theorem	p. 226
6.2 Graphs	p. 238
6.3 Manifolds in Parametric Representations	p. 243
6.4 Manifolds in Implicit Representations	p. 252
6.5 Differentiation on Manifolds	p. 260
7 Higher-Order Derivatives	p. 267
7.1 Definitions	p. 267
7.2 Change of Order in Differentiation	p. 270
7.3 Sequences of Polynomials	p. 273
7.4 Local Extremal Values	p. 282
Part III Integration
8 Multiple Integrals	p. 287
8.1 Jordan Sets and Volume	p. 289
8.2 Integrals	p. 303
8.3 Images of Jordan Sets	p. 321
8.4 Change of Variables	p. 328
9 Integration on Manifolds	p. 339
9.1 Euclidean Volumes	p. 340
9.2 Integration on Manifolds	p. 345
9.3 Oriented Manifolds	p. 353
9.4 Integrals of Vector Fields	p. 361
9.5 Integrals of Tensor Fields	p. 366
9.6 Integration on Graphs	p. 371
10 Stokes' Theorem	p. 381
10.1 Basic Stokes' Theorem	p. 382
10.2 Flows	p. 386
10.3 Flux and Change of Volume in a Flow	p. 390
10.4 Exterior Derivatives	p. 396
10.5 Regular and Almost Regular Sets	p. 401
10.6 Stokes' theorem on Manifolds	p. 412
Part IV Appendices
Appendix A Construction of the real numbers	p. 419
A.1 Field and Order Axioms in Q	p. 420
A.2 Equivalence Classes of Cauchy Sequences in Q	p. 421
A.3 Completeness of R	p. 427
Appendix B Dimension of a vector space	p. 431
B.1 Bases and linearly independent subsets	p. 432
Appendix C Determinants	p. 435
C.1 Permutations	p. 435
C.2 Determinants of Square Matrices	p. 437
C.3 Determinant Functions	p. 439
C.4 Determinant of a Linear Transformation	p. 443
C.5 Determinants on Cartesian Products	p. 444
C.6 Determinants in Euclidean Spaces	p. 445
C.7 Trace of an Operator	p. 448
Appendix D Partitions of unity	p. 451
D.1 Partitions of Unity	p. 452
Index	p. 455

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