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This page intentionally left blank ANALYSIS IN VECTOR SPACES This page intentionally left blank ANALYSIS IN VECTOR SPACES A Course in Advanced Calculus Mustafa A. Akcoglu University of Toronto Paul F. A. Bartha University of British Columbia Dzung M. Ha Ryerson University WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION Copyright ©2009 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. 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Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data: Analysis in Vector Spaces / Mustafa Akcoglu, Paul Bartha, Dzung Minh Ha. Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 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 Completenes 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 Conected Sets 54 v VI CONTENTS 3 Vector Functions 61 3.1 Vector Spaces: The Basics 62 3.2 Bilinear Functions 82 3.3 Multilinear Functions 8 3.4 Iner Products 95 3.5 Orthogonal Projections 103 3.6 Spectral Theorem 109 PART I 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 Diferentiable Functions 190 5.3 Existence of Derivatives 201 5.4 Partial Derivatives 205 5.5 Rules of Diferentiation 21 5.6 Diferentiation of Products 218 6 Difeomorphisms and Manifolds 25 6.1 The Inverse Function Theorem 26 6.2 Graphs 238 6.3 Manifolds in Parametric Representations 243 6.4 Manifolds in Implicit Representations 252 6.5 Diferentiation on Manifolds 260 7 Higher-Order Derivatives 267 7.1 Definitions 267 7.2 Change of Order in Diferentiation 270 7.3 Sequences of Polynomials 273 CONTENTS VI 7.4 Local Extremal Values 282 PART I 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 39 9.1 Euclidean Volumes 340 9.2 Integration on Manifolds 345 9.3 Oriented Manifolds 353 9.4 Integrals of Vector Fields 361 9.5 Integrals of Tensor Fields 36 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 Apendix A: Construction of the real numbers 419 A. 1 Field and Order Axioms in Q 420 A.2 Equivalence Clases of Cauchy Sequences in Q 421 A.3 Completenes of R 427 Apendix B: Dimension of a vector space 431 B.l Bases and linearly independent subsets 432 Appendix C: Determinants 435 VÜi CONTENTS C.l Permutations 435 C.2 Determinants of Square Matrices 437 C.3 Determinant Functions 439 C.4 Determinant of a Linear Transformation 43 C.5 Determinants on Cartesian Products 4 C.6 Determinants in Euclidean Spaces 45 C.7 Trace of an Operator 48 Apendix D: Partitions of unity 451 D.l Partitions of Unity 452 Index 455