Analysis in Vector Spaces

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  • Format: Hardcover
  • Copyright: 2009-02-03
  • Publisher: Wiley

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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.

Author Biography

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.

Table of Contents

Prefacep. ix
Background Material
Sets and Functionsp. 3
Sets in Generalp. 3
Sets of Numbersp. 10
Functionsp. 17
Real Numbersp. 31
Review of the Order Relationsp. 32
Completeness of Real Numbersp. 36
Sequences of Real Numbersp. 40
Subsequencesp. 45
Series of Real Numbersp. 50
Intervals and Connected Setsp. 54
Vector Functionsp. 61
Vector Spaces: The Basicsp. 62
Bilinear Functionsp. 82
Multilinear Functionsp. 88
Inner Productsp. 95
Orthogonal Projectionsp. 103
Spectral Theoremp. 109
Normed Vector Spacesp. 123
Preliminariesp. 124
Convergence in Normed Spacesp. 128
Norms of Linear and Multilinear Transformationsp. 135
Continuity in Normed Spacesp. 142
Topology of Normed Spacesp. 156
Derivativesp. 175
Functions of a Real Variablep. 176
Differentiable Functionsp. 190
Existence of Derivativesp. 201
Partial Derivativesp. 205
Rules of Differentiationp. 211
Differentiation of Productsp. 218
Diffeomorphisms and Manifoldsp. 225
The Inverse Function Theoremp. 226
Graphsp. 238
Manifolds in Parametric Representationsp. 243
Manifolds in Implicit Representationsp. 252
Differentiation on Manifoldsp. 260
Higher-Order Derivativesp. 267
Definitionsp. 267
Change of Order in Differentiationp. 270
Sequences of Polynomialsp. 273
Local Extremal Valuesp. 282
Multiple Integralsp. 287
Jordan Sets and Volumep. 289
Integralsp. 303
Images of Jordan Setsp. 321
Change of Variablesp. 328
Integration on Manifoldsp. 339
Euclidean Volumesp. 340
Integration on Manifoldsp. 345
Oriented Manifoldsp. 353
Integrals of Vector Fieldsp. 361
Integrals of Tensor Fieldsp. 366
Integration on Graphsp. 371
Stokes' Theoremp. 381
Basic Stokes' Theoremp. 382
Flowsp. 386
Flux and Change of Volume in a Flowp. 390
Exterior Derivativesp. 396
Regular and Almost Regular Setsp. 401
Stokes' theorem on Manifoldsp. 412
Construction of the real numbersp. 419
Field and Order Axioms in Qp. 420
Equivalence Classes of Cauchy Sequences in Qp. 421
Completeness of Rp. 427
Dimension of a vector spacep. 431
Bases and linearly independent subsetsp. 432
Determinantsp. 435
Permutationsp. 435
Determinants of Square Matricesp. 437
Determinant Functionsp. 439
Determinant of a Linear Transformationp. 443
Determinants on Cartesian Productsp. 444
Determinants in Euclidean Spacesp. 445
Trace of an Operatorp. 448
Partitions of unityp. 451
Partitions of Unityp. 452
Indexp. 455
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