Analysis in Vector Spaces / Edition 1

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A Rigorous Introdution To Calculus In Vector Spaces
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Editorial Reviews

From the Publisher
"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." (CHOICE, November 2010)
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Product Details

  • ISBN-13: 9780470148242
  • Publisher: Wiley, John & Sons, Incorporated
  • Publication date: 2/9/2009
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 480
  • Product dimensions: 6.30 (w) x 9.30 (h) x 1.10 (d)

Meet the Author

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.

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Table of Contents

Preface ix

Part I Background Material

1 Sets and Functions 3

1.1 Sets in General 3

1.2 Sets of Numbers 10

1.3 Functions 17

2 Real Numbers 31

2.1 Review of the Order Relations 32

2.2 Completeness of Real Numbers 36

2.3 Sequences of Real Numbers 40

2.4 Subsequences 45

2.5 Series of Real Numbers 50

2.6 Intervals and Connected Sets 54

3 Vector Functions 61

3.1 Vector Spaces: The Basics 62

3.2 Bilinear Functions 82

3.3 Multilinear Functions 88

3.4 Inner Products 95

3.5 Orthogonal Projections 103

3.6 Spectral Theorem 109

Part II Differentiation

4 Normed Vector Spaces 123

4.1 Preliminaries 124

4.2 Convergence in Normed Spaces 128

4.3 Norms of Linear and Multilinear Transformations 135

4.4 Continuity in Normed Spaces 142

4.5 Topology of Normed Spaces 156

5 Derivatives 175

5.1 Functions of a Real Variable 176

5.2 Differentiable Functions 190

5.3 Existence of Derivatives 201

5.4 Partial Derivatives 205

5.5 Rules of Differentiation 211

5.6 Differentiation of Products 218

6 Diffeomorphisms and Manifolds 225

6.1 The Inverse Function Theorem 226

6.2 Graphs 238

6.3 Manifolds in Parametric Representations 243

6.4 Manifolds in Implicit Representations 252

6.5 Differentiation on Manifolds 260

7 Higher-Order Derivatives 267

7.1 Definitions 267

7.2 Change of Order in Differentiation 270

7.3 Sequences of Polynomials 273

7.4 Local Extremal Values 282

Part III Integration

8 Multiple Integrals 287

8.1 Jordan Sets and Volume 289

8.2 Integrals 303

8.3 Images of Jordan Sets 321

8.4 Change of Variables 328

9 Integration on Manifolds 339

9.1 Euclidean Volumes 340

9.2 Integration on Manifolds 345

9.3Oriented Manifolds 353

9.4 Integrals of Vector Fields 361

9.5 Integrals of Tensor Fields 366

9.6 Integration on Graphs 371

10 Stokes' Theorem 381

10.1 Basic Stokes' Theorem 382

10.2 Flows 386

10.3 Flux and Change of Volume in a Flow 390

10.4 Exterior Derivatives 396

10.5 Regular and Almost Regular Sets 401

10.6 Stokes' theorem on Manifolds 412

part IV Appendices

Appendix A Construction of the real numbers 419

A.1 Field and Order Axioms in Q 420

A.2 Equivalence Classes of Cauchy Sequences in Q 421

A.3 Completeness of R 427

tAppendix B Dimension of a vector space 431

B.1 Bases and linearly independent subsets 432

Appendix C Determinants 435

C.1 Permutations 435

C.2 Determinants of Square Matrices 437

C.3 Determinant Functions 439

C.4 Determinant of a Linear Transformation 443

C.5 Determinants on Cartesian Products 444

C.6 Determinants in Euclidean Spaces 445

C.7 Trace of an Operator 448

Appendix D Partitions of unity 451

D.1 Partitions of Unity 452

Index 455

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