Analysis in Vector Spaces

This text is intended for a second course in analysis, concentrating on differentiation and integration of functions of several variables. The material is presented as a unified course on vector spaces. The book successfully provides a firm mathematical foundation for further study and established an indispensable background for further studies in mathematics or related disciplines that involve heavy use of mathematical techniques and concepts. The book also provides exposure to a variety of other mathematical topics such as basic point-set topology, measure theory, differential geometry, and the theory of manifolds.

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 and operator theory.

Preface	p. ix
Background Material
Sets and Functions	p. 3
Sets in General	p. 3
Sets of Numbers	p. 10
Functions	p. 17
Real Numbers	p. 31
Review of the Order Relations	p. 32
Completeness of Real Numbers	p. 36
Sequences of Real Numbers	p. 40
Subsequences	p. 45
Series of Real Numbers	p. 50
Intervals and Connected Sets	p. 54
Vector Functions	p. 61
Vector Spaces: The Basics	p. 62
Bilinear Functions	p. 82
Multilinear Functions	p. 88
Inner Products	p. 95
Orthogonal Projections	p. 103
Spectral Theorem	p. 109
Differentiation
Normed Vector Spaces	p. 123
Preliminaries	p. 124
Convergence in Normed Spaces	p. 128
Norms of Linear and Multilinear Transformations	p. 135
Continuity in Normed Spaces	p. 142
Topology of Normed Spaces	p. 156
Derivatives	p. 175
Functions of a Real Variable	p. 176
Differentiable Functions	p. 190
Existence of Derivatives	p. 201
Partial Derivatives	p. 205
Rules of Differentiation	p. 211
Differentiation of Products	p. 218
Diffeomorphisms and Manifolds	p. 225
The Inverse Function Theorem	p. 226
Graphs	p. 238
Manifolds in Parametric Representations	p. 243
Manifolds in Implicit Representations	p. 252
Differentiation on Manifolds	p. 260
Higher-Order Derivatives	p. 267
Definitions	p. 267
Change of Order in Differentiation	p. 270
Sequences of Polynomials	p. 273
Local Extremal Values	p. 282
Integration
Multiple Integrals	p. 287
Jordan Sets and Volume	p. 289
Integrals	p. 303
Images of Jordan Sets	p. 321
Change of Variables	p. 328
Integration on Manifolds	p. 339
Euclidean Volumes	p. 340
Integration on Manifolds	p. 345
Oriented Manifolds	p. 353
Integrals of Vector Fields	p. 361
Integrals of Tensor Fields	p. 366
Integration on Graphs	p. 371
Stokes' Theorem	p. 381
Basic Stokes' Theorem	p. 382
Flows	p. 386
Flux and Change of Volume in a Flow	p. 390
Exterior Derivatives	p. 396
Regular and Almost Regular Sets	p. 401
Stokes' theorem on Manifolds	p. 412
Appendices
Construction of the real numbers	p. 419
Field and Order Axioms in Q	p. 420
Equivalence Classes of Cauchy Sequences in Q	p. 421
Completeness of R	p. 427
Dimension of a vector space	p. 431
Bases and linearly independent subsets	p. 432
Determinants	p. 435
Permutations	p. 435
Determinants of Square Matrices	p. 437
Determinant Functions	p. 439
Determinant of a Linear Transformation	p. 443
Determinants on Cartesian Products	p. 444
Determinants in Euclidean Spaces	p. 445
Trace of an Operator	p. 448
Partitions of unity	p. 451
Partitions of Unity	p. 452
Index	p. 455
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Analysis in Vector Spaces

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