A Companion to Analysis A Second First and First Second Course in Analysis

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A Companion to Analysis A Second First and First Second Course in Analysis

T. W. Korner

Graduate Studies

in Mathematics

Volume 62

a 5 American Mathematical Society Providence, Rhode Island

E D I T O R I A L C O M M I T T E E

Walter Craig Nikolai Ivanov

Steven G. Krantz David Saltman (Chair)

2000 Mathematics Subject Classification. Primary 26-01.

For additional information and updates on this book, visit www.ams.org/bookpages /gsm-62

Library of Congress Cataloging-in-Publication D a t a

Korner, T. W. (Thomas Williamm), 1946-A companion to analysis : a second first and first second course in analysis / T. W. Korner.

p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 62) Includes bibliographical references and index. ISBN 0-8218-3447-9 (alk. paper) 1. Mathematical analysis. I. Title. II. Series.

QA300.K589 2003 515—dc22 2003062905

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10 9 8 7 6 5 4 3 2 1 09 08 07 06 05 04

[Archimedes] concentrated his ambition exclusively upon those specu­lations which are untainted by the claims of necessity. These studies, he believed, are incomparably superior to any others, since here the grandeur and beauty of the subject matter vie for our admiration with the cogency and precision of the methods of proof. Certainly in the whole science of ge­ometry it is impossible to find more difficult and intricate problems handled in simpler and purer terms than in his works. Some writers attribute it to his natural genius. Others maintain that phenomenal industry lay behind the apparently effortless ease with which he obtained his results. The fact is that no amount of mental effort of his own would enable a man to hit upon the proof of one of Archimedes' theorems, and yet as soon as it is explained to him, he feels he might have discovered it himself, so smooth and rapid is the path by which he leads us to the required conclusion.

Plutarch Life of Marcellus [Scott-Kilvert's translation]

It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible argument, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, choosing rather to acknowledge their ignorance, than affirm anything rashly.

Barrow Mathematical Lectures

For [A. N.] Kolmogorov mathematics always remained in part a sport. But when . . . I compared him with a mountain climber who made first ascents, contrasting him with I. M. Geffand whose work I compared with the building of highways, both men were offended. ' . . . Why, you don't think I am capable of creating general theories?' said Andrei Nikolaevich. 'Why, you think I can't solve difficult problems?' added I. M.

V. I. Arnol'd in Kolmogorov in Perspective

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Contents

Introduction

Chapter 1. The Real Line

§1.1. Why do we bother?

§1.2. Limits

§1.3. Continuity

§1.4. The fundamental axiom

§1.5. The axiom of Archimedes

§1.6. Lion hunting

§1.7. The mean value inequality

§1.8. Full circle

§1.9. Are the real numbers unique?

Chapter 2. A First Philosophical Interlude W

§2.1. Is the intermediate value theorem obvious? W

Chapter 3. Other Versions of the Fundamental Axiom

§3.1. The supremum

§3.2. The Bolzano-Weierstrass theorem

§3.3. Some general remarks

vm Contents

Chapter 4. Higher Dimensions 43

§4.1. Bolzano-Weierstrass in Higher Dimensions 43

§4.2. Open and closed sets 48

§4.3. A central theorem of analysis 56

§4.4. The mean value theorem 59

§4.5. Uniform continuity 64

§4.6. The general principle of convergence 66

Chapter 5. Sums and Suchlike V 73

§5.1. Comparison tests W 73

§5.2. Conditional convergence <s? 75

§5.3. Interchanging limits 9 80

§5.4. The exponential function <s? 88

§5.5. The trigonometric functions 9 95

§5.6. The logarithm V 99

§5.7. Powers 9 105

§5.8. The fundamental theorem of algebra 9 109

Chapter 6. Differentiation 117

§6.1. Preliminaries 117

§6.2. The operator norm and the chain rule 123

§6.3. The mean value inequality in higher dimensions 130

Chapter 7. Local Taylor Theorems 135

§7.1. Some one-dimensional Taylor theorems 135

§7.2. Some many-dimensional local Taylor theorems 139

§7.3. Critical points 147

Chapter 8. The Riemann Integral 161

§8.1. Where is the problem ? 161

§8.2. Riemann integration 164

§8.3. Integrals of continuous functions 173

§8.4. First steps in the calculus of variations 9 181

§8.5. Vector-valued integrals 192

Contents IX

Chapter 9. Developments and Limitations of the Riemann Integral 9 195

§9.1. Why go further? 195

§9.2. Improper integrals 9 197

§9.3. Integrals over areas 9 201

§9.4. The Riemann-Stieltjes integral 9 206

§9.5. How long is a piece of string? 9 213

Chapter 10. Metric Spaces 221

§10.1. Sphere packing 9 221

§10.2. Shannon's theorem V 224

§10.3. Metric spaces 229

§10.4. Norms and the interaction of algebra and analysis 234

§10.5. Geodesies 9 241

Chapter 11. Complete Metric Spaces 249

§11.1. Completeness 249

§11.2. The Bolzano-Weierstrass property 257

§11.3. The uniform norm 261

§11.4. Uniform convergence 265

§11.5. Power series 273

§11.6. Fourier series 9 282

Chapter 12. Contraction Mappings and Differential Equations 287

§12.1. Banach's contraction mapping theorem 287

§12.2. Existence of solutions of differential equations 289

§12.3. Local to global 9 294

§12.4. Green's function solutions 9 301

Chapter 13. Inverse and Implicit Functions 311

§13.1. The inverse function theorem 311

§13.2. The implicit function theorem 9 320

§13.3. Lagrange multipliers 9 328

X Contents

Chapter 14. Completion 335

§14.1. What is the correct question? 335

§14.2. The solution 341

§14.3. Why do we construct the reals? 9 344

§14.4. How do we construct the reals? 9 348

§14.5. Paradise lost? W 354

Appendix A. Ordered Fields 357

Appendix B. Countability 361

Appendix C. The Care and Treatment of Counterexamples 365

Appendix D. A More General View of Limits 371

Appendix E. Traditional Partial Derivatives 377

Appendix F. Another Approach to the Inverse Function Theorem 383

Appendix G. Completing Ordered Fields 387

Appendix H. Constructive Analysis 391

Appendix I. Miscellany 395

Appendix J. Executive Summary 401

Appendix K. Exercises 405

Bibliography 583

Index 585

Introduction

In his autobiography [12], Winston Churchill remembered his struggles with Latin at school. ' . . . even as a schoolboy I questioned the aptness of the Classics for the prime structure of our education. So they told me how Mr Gladstone read Homer for fun, which I thought served him right.' 'Naturally' he says 'I am in favour of boys learning English. I would make them all learn English; and then I would let the clever ones learn Latin as an honour, and Greek as a treat.'

This book is intended for those students who might find rigorous anal­ysis a treat. The content of this book is summarised in Appendix J and corresponds more or less (more rather than less) to a recap at a higher level of the first course in analysis followed by the second course in analysis at Cambridge in 2004 together with some material from various methods courses (and thus corresponds to about 60 to 70 hours of lectures). Like those courses, it aims to provide a foundation for later courses in functional analysis, differential geometry and measure theory. Like those courses also, it assumes complementary courses such as those in mathematical methods and in elementary probability to show the practical uses of calculus and strengthen computational and manipulative skills. In theory, it starts more or less from scratch, but the reader who finds the discussion of Section 1.1 baffling or the e, 5 arguments of Section 1.2 novel will probably find this book unrewarding. I assume a fair degree of algebraic fluency and, from Chapter 4 onwards, some exposure to linear algebra.

This book is about mathematics for its own sake. It is a guided tour of a great but empty Opera House. The guide is enthusiastic but interested only in sight-lines, acoustics, lighting and stage machinery. If you wish to see the

XI

X l l Introduction

stage filled with spectacle and the air filled with music you must come at another time and with a different guide.

Although I hope this book may be useful to others, I wrote it for students to read either before or after attending the appropriate lectures. For this reason, I have tried to move as rapidly as possible to the points of difficulty, show why they are points of difficulty and explain clearly how they are overcome. If you understand the hardest part of a course then, almost automatically, you will understand the easiest. The converse is not true.

In order to concentrate on the main matter in hand, some of the simpler arguments have been relegated to exercises. The student reading this book before taking the appropriate course may take these results on trust and concentrate on the central arguments which are given in detail. The stu­dent reading this book after taking the appropriate course should have no difficulty with these minor matters and can also concentrate on the central arguments. I think that doing at least some of the exercises will help stu­dents to 'internalise' the material, but I hope that even students who skip most of the exercises can profit from the rest of the book.

I have included further exercises in Appendix K. Some are standard, some form commentaries on the main text and others have been taken or adapted from the Cambridge mathematics exams. None are 'makeweights', they are all intended to have some point of interest. Sketches of some solu­tions are available from the home pages given on page xiii. I have tried to keep to standard notations, but a couple of notational points are mentioned in the index under the heading 'notation'.

I have not tried to strip the subject down to its bare bones. A skeleton is meaningless unless one has some idea of the being it supports, and that being in turn gains much of its significance from its interaction with other beings, both of its own species and of other species. For this reason, I have included several sections marked by a ^?. These contain material which is not necessary to the main argument but which sheds light on it. Ideally, the student should read them but not study them with anything like the same attention which she devotes to the unmarked sections. There are two sections marked W which contain some, very simple, philosophical discussion. It is entirely intentional that removing the appendices and the sections marked with a <? more than halves the length of the book.

It is an honour to publish with the AMS; I am grateful to the editorial staff for making it a pleasure. I thank Dr Gunther Leobacher for com­puter generating the figures for this book. I owe particular thanks to Jorge Aarao, Brian Blank, Johan Grundberg, Jonathan Partington, Ralph Sizer, Thomas Ward and an anonymous referee, but I am deeply grateful to the many other people who pointed out errors in and suggested improvements

Introduction xin

to earlier versions of this book. My e-mail address is twk@dpmms. cam. ac .uk and I shall try to keep a list of corrections accessible from an AMS page at www.ams.org/bookpages/gsm-62 as well as from my home page page at http://www.dpmms.cam.ac.uk/~twk/.

I learned calculus from the excellent Calculus [13] of D. R. Dickinson and its inspiring author. My first glimpse of analysis was in Hardy's Pure Math­ematics [24] read when I was too young to really understand it. I learned elementary analysis from Ferrar's A Textbook of Convergence [18] (an ex­cellent book for those making the transition from school to university, now, unfortunately, out of print) and Burkill's A First Course in Mathematical Analysis [10]. The books of Kolmogorov and Fomin [31] and, particularly, Dieudonne [14] showed me that analysis is not a collection of theorems but a single coherent theory. Stromberg's book An Introduction to Classical Real Analysis [48] lies permanently on my desk for browsing. The expert will easily be able to trace the influence of these books on the pages that follow. If, in turn, my book gives any student half the pleasure that the ones just cited gave me, I will feel well repaid.

Cauchy began the journey that led to the modern analysis course in his lectures at the Ecole Polytechnique in the 1820's. The times were not propitious. A reactionary government was determined to keep close control over the school. The faculty was divided along fault lines of politics, religion and age whilst physicists, engineers and mathematicians fought over the contents of the courses. The student body arrived insufficiently prepared and then divided its time between radical politics and worrying about the job market (grim for both staff and students). Cauchy's course was not popular1.

Everybody can sympathise with Cauchy's students who just wanted to pass their exams and with his colleagues who just wanted the standard material taught in the standard way. Most people neither need nor want to know about rigorous analysis. But there remains a small group for whom the ideas and methods of rigorous analysis represent one of the most splendid triumphs of the human intellect. We echo Cauchy's defiant preface to his printed lecture notes.

As to the methods [used here], I have sought to endow them with all the rigour that is required in geometry and in such a way that I have not had to have recourse to formal ma­nipulations. Such arguments, although commonly accepted . . . cannot be considered, it seems to me, as anything other than [suggestive] to be used sometimes in guessing the truth.

Belhoste's splendid biography [4] gives the fascinating details.

XIV Introduction

Such reasons [moreover] ill agree with the mathematical sci­ences' much vaunted claims of exactitude. It should also be observed that they tend to attribute an indefinite extent to algebraic formulas when, in fact, these formulas hold under certain conditions and for only certain values of the variables involved. In determining these conditions and these values and in settling in a precise manner the sense of the notation and the symbols I use, I eliminate all uncertainty. . . . It is true that in order to remain faithful to these principles, I sometimes find myself forced to depend on several proposi­tions that perhaps seem a little hard on first encounter . . . . But, those who will read them will find, I hope, that such propositions, implying the pleasant necessity of endowing the theorems with a greater degree of precision and restricting statements which have become too broadly extended, will actually benefit analysis and will also provide a number of topics for research, which are surely not without importance.

I dedicate this book to the memory of my parents in gratitude for many years of love and laughter.

Bibliography

1. F. S. Acton, Numerical methods that work, Harper and Row, 1970.

2. A. F. Beardon, Limits, a new approach to real analysis, Springer, 1997.

3. R. Beigel, Irrationality without number theory, American Mathematical Monthly 98 (1991), 332-335.

4. B. Belhoste, Augustin-Louis Cauchy. a biography, Springer, 1991, Translated from the French by F. Ragland, but the earlier French publication of the same author is a different book.

5. D. Berlinski, A tour of the calculus, Pantheon Books, New York, 1995.

6. P. Billingsley, Probability and measure, Wiley, 1979.

7. E. Bishop and D. Bridges, Constructive analysis, Springer, 1985.

8. R. P. Boas, Lion hunting and other mathematical pursuits, Dolciani Mathematical Expositions, vol. 15, MAA, 1995.

9. J. R. Brown, Philosophy of mathematics, Routledge and Kegan Paul, 1999.

10. J. C. Burkill, A first course in mathematical analysis, CUP, 1962.

11. R. P. Burn, Numbers and functions, CUP, 1992.

12. W. S. Churchill, My early life, Thornton Butterworth, London, 1930.

13. D. R. Dickinson, Calculus, George G. Harrap and Co., London, 1958.

14. J. Dieudonne, Foundations of modern analysis, Academic Press, 1960.

15. , Infinitesimal calculus, Kershaw Publishing Company, London, 1973, Trans­lated from the French Calcul Infinitesimal published by Hermann, Paris in 1968.

16. R. M. Dudley, Real analysis and probability, second ed., Wadsworth and Brooks, CUP, 2002.

17. J. Fauvel and J. Gray (eds.), History of mathematics: A reader, Macmillan, 1987.

18. W. L. Ferrar, A textbook of convergence, OUP, 1938.

19. J. E. Gordon, Structures, or why things don't fall down, Penguin, 1978.

20. A. Griinbaum, Modern science and Zeno's paradoxes, George Allen and Unwin, 1968.

21. P. R. Halmos, Naive set theory, Van Nostrand, 1960.

583

584 Bibliography

22. , / want to be a mathematician, Springer, 1985.

23. G. H. Hardy, Collected papers, vol. V, OUP, 1908.

24. , A course of pure mathematics, CUP, 1908, Still available in its 10th edition.

25. , Divergent series, OUP, 1949.

26. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, CUP, 1938.

27. P. T. Johnstone, Notes on logic and set theory, CUP, 1987.

28. Y. Katznelson and K. Stromberg, Everywhere differentiate, nowhere monotone, func­tions, American Mathematical Monthly 81 (1974), 349-54.

29. F. Klein, Elementary mathematics from an advanced standpoint (part 1), Dover, 1924, Third edition. Translated by E. R. Hedrick and C. A. Hedrick.

30. M. Kline, Mathematical thought from ancient to modern times, OUP, 1972.

31. A. N. Kolmogorov and S. V. Fomin, Introductory real analysis, Prentice Hall, 1970, Translated from the Russian and edited by R. A. Silverman.

32. T.W. Korner, The behavior of power series on their circle of convergence, Banach Spaces, Harmonic Analysis, and Probability Theory, Lecture Notes in Mathematics, vol. 995, Springer, 1983, pp. 56-94.

33. , Differentiable functions on the rationals, Bulletin of the London Mathematical Society 23 (1991), 557-62.

34. , Butterfly hunting and the symmetry of mixed partial derivatives, American Mathematical Monthly 105 (1998), 756-8.

35. E. Laithwaite, Invitation to engineering, Blackwell, 1984.

36. J. E. Littlewood, A mathematician's miscellany, 2nd ed., CUP, 1985, (Editor B. Bol-lobas).

37. P. Mancosu and E. Vilati, TorricellVs infinitely long horn and its philosophical recep­tion in the seventeeenth century, Isis 82 (1991), 50-70.

38. J. C. Maxwell, Treatise on electricity and magnetism, OUP, 1873.

39. , Scientific letters and papers, vol. II, CUP, 1995.

40. D. J. Newman, A problem seminar, Springer, 1982.

41. G. M. Phillips, Two millennia of mathematics. from Archimedes to Gauss, CMS books in mathematics 6, Springer, 2000.

42. Plato, Parmenides, Routledge and Kegan Paul, 1939, Translated by F. M. Cornford.

43. F. Poundstone, Labyrinths of reason, Penguin, 1991.

44. M. J. D. Powell, Approximation theory and methods, CUP, 1988.

45. T. A. Ptaclusp, Episodes from the lives of the great accountants, Pyramid Press, Tsort, 1970.

46. M. C. Reed, Fundamental ideas of analysis, Wiley, 1998.

47. P. L. Roe, The best shape for a tin can, The Mathematical Gazette 75 (1991), no. 472, 147-50.

48. K. R. Stromberg, Introduction to classical real analysis, Wadsworth, 1981.

49. S. Wagon, The Banach-Tarski paradox, CUP, 1985.

50. E. T. Whittaker and G. N. Watson, A course of modem analysis, CUP, 1902, Still available in its 4th edition.

51. P. Whittle, Optimization under constraints, Wiley, 1971.

Index

You may find Appendix J useful

absolute value, 358 absurd absurdums, 449 abuse of language, 396 algebraically closed, 116 algebraists, dislike metrics, 123, 568 alternating series test, 76 antiderivative, existence and uniqueness, 177 area, general problems, 161-164, 203-206,

218-220 authors, other

Beardon, 56, 372 Berlinski, 20 Billingsly, 218 Boas, 59, 145, 201 Bourbaki, 355, 396 Burn, 62 Conway, 422, 449 Dieudonne, xiii, 25, 59, 147, 196 Halmos, 355, 369 Hardy, xiii, 43, 81, 100, 147, 281 Klein, 109, 396 Kline, 354 Littlewood, 79 Petard, H., 16 Plato, 28 Poincare, 355

axiom fundamental, 9, 12, 21, 353 of Archimedes, 10, 12, 352, 388 of choice, 164, 239

axioms for a metric space, 230 for an ordered field, 357 general discussion, 230, 344-345, 355-356 Zermelo-Fraenkel, 355

tracking down particular ideas.

balls open and closed, 50, 232 packing, 221-223 packing in F%, 225-226

Banach, 230, 287 Bernstein polynomial, 521 Big Oh and little oh, 483 bijective function, 449 binomial expansion

for general exponent, 280 positive integral exponent, 543

binomial theorem, 280, 543 bisection, bisection search, see lion hunting Bishop's constructive analysis, 391-394 Bolzano-Weierstrass

and compactness, 395 and sequential compactness, 258 and total boundedness, 259 equivalent to fundamental axiom, 40 for closed bounded sets in Mm , 49 for metric spaces, 258-260 for R, 37-39 in E m , 47

bounded variation, functions of, 172, 492-495, 579

brachistochrone problem, 181

calculus of variations problems, 189-192 successes, 181-189 used, 245

Cantor set, 533 Cauchy

condensation test, 74 father of modern analysis, xiii, 544 function not given by Taylor series, 137

585

586 Index

mean value theorem, 430 proof of binomial theorem, 543 sequence, 66, 249 solution of differential equations, 544

Cauchy-Riemann equations, 455 Cauchy-Schwarz inequality, 44 Cayley-Hamilton theorem, 568 chain rule

many-dimensional, 126-127 one-dimensional, 99-100

Chebyshev, see Tchebychev chestnuts, old, 179, 409, 441, 450, 462 chords, 450 closed bounded sets in Rm

and Bolzano-Weierstrass, 49 and continuous functions, 56-57, 65 compact, 395 nested, 59

closed sets complement of open sets, 50, 233 definition for metric space, 232 definition for E m , 48 key properties, 51, 233

closure algebraic, 116 metric, 503

comma notation, 122, 141 compactness, 395, 515 completeness

definition, 249 proving completeness, 253 proving incompleteness, 250

completion discussion, 335—338 existence, 341-344, 576 ordered fields, 387-389 structure carries over, 338-341 unique, 336-338

constant value theorem false for rationals, 2 many-dimensional, 133 true for reals, 20

construction of C from R, 347-348 Q from Z, 346-347 R from Q, 348-354 Z from N, 346

continued fractions, 411-414 continuity, see also uniform continuity

discussed, 7, 366-369, 393 of linear maps, 124, 238-240 pointwise, 7, 53, 233 via open sets, 53, 234

continuous functions exotic, 2, 531-535 integration of, 173-177 on closed bounded sets in Mm, 56-58

continuum, models for, see reals and ratio­nals, 25-28, 394

contraction mapping, 287-289, 291, 313, 384 convergence tests for sums

Abel's, 77, 442 alternating series, 76 Cauchy condensation, 74 comparison, 69 discussion of, 438 integral comparison, 198 ratio, 74

convergence, pointwise and uniform, 265 convex

function, 424, 474-476 set, 420, 551

convolution, 546-547 count ability, 361-364 critical points, see also maxima and minima,

147-153, 156-160, 321

D notation, 122, 144 Darboux, theorems of, 425, 469 decimal expansion, 13 delta function, 210, 302 dense sets as skeletons, 12, 336, 522 derivative

complex, 273-274 directional, 121 general discussion, 117-123 in applied mathematics, 145, 377-380 in many dimensions, 120 in one dimension, 18 left and right, 398 more general, 241 not continuous, 425 partial, 122

devil's staircase, 533 diagrams, use of, 95 differential equations

and Green's functions, 301-308 and power series, 278, 544 Euler's method, 557-560 existence and uniqueness of solutions, 289-

301 differentiation

Fourier series, 286 power series, 276, 537-539 term by term, 276 under the integral

finite range, 182, 527 infinite range, 272, 526

Dini's theorem, 521 directed set, 372 dissection, 164 dominated convergence

for some integrals, 527 for sums, 82

Index 587

duck, tests for, 348

economics, fundamental problem of, 57 escape to infinity, 82, 268-269 Euclidean

geometry, 344-345 norm, 44

Euler method for differential equations, 557-560 on homogeneous functions, 456

Euler's 7, 441 Euler-Lagrange equation, 185 exponential function, 89-95, 137, 300, 393,

474, 571 extreme points, 420-421

Father Christmas, 164 fixed point theorems, 17, 287-289 Fourier series, 282-286 Fubini's theorem

for infinite integrals, 488 for integrals of continuous functions, 202,

486 for sums, 87

full rank, 327 functional equations, 453-455 fundamental axiom, 9 fundamental theorem of algebra

proof, 110-113 statement, 109 theorem of analysis, 109, 115

fundamental theorem of the calculus discussion of extensions, 177 in one dimension, 175-177

Gabriel's horn, 498 Gaussian quadrature, 523 general principle

of convergence, 67, 249, 388 of uniform convergence, 266

generic, 157 geodesies, 241-247 global and local, contrasted, 64, 119, 136-

138, 149, 153, 157, 297-300, 323 Greek rigour, 29, 345, 355, 498 Green's functions, 301-309, 562-566

Hahn-Banach for Rn , 420 Hausdorff metric, 513-514 Heaviside step function, 206 Heine-Borel theorem, 422 Hessian, 150 hill and dale theorem, 157-159 Holder's inequality, 509, 511 homeomorphism, 580 homogeneous function, 456

identity of mathematical objects, 348 implicit function theorem

discussion, 320-328 statement and proof, 324-325

indices, see powers inequality

arithmetic-geometric, 425 Cauchy-Schwarz, 44 Holder's, 509, 511 Jensen's, 424, 474 Ptolomey's, 416 reverse Holder, 510 Tchebychev, 211

infimum, 34 infinite

products, 446, 542 sums, see sums

injective function, 449 inner product

completion, 340, 343 for/2 , 508 for W1, 43

integral kernel, example of, 309 integral mean value theorem, 467 integrals

along curves, 216-218, 220 and uniform convergence, 267 improper (or infinite), 197-201 of continuous functions, 173-177 over area, 201-206 principle value, 201 Riemann, definition, 164-166 Riemann, problems, 195-196, 203 Riemann, properties, 166-173 Riemann-Stieltjes, 206-213, 495 Riemann-Stieltjes, problems, 209 vector-valued, 192-194

integration by parts, 180 by substitution, 178 numerical, 471-473, 523 Riemann versus Lebesgue, 196-197 term by term, 272

interchange of limits derivative and infinite integral, 272, 526 derivative and integral, 182, 527 general discussion, 80-82 infinite integrals, 488 integral and sum, 272 integrals, 202 limit and derivative, 270, 271, 526 limit and integral, 267-269 limit and sum, 82 partial derivatives, 143 sums, 87

interior of a set, 503 interlaced zeros, 525, 561

588 Index

intermediate value theorem equivalent to fundamental axiom, 21 false for rationals, 2 not available in constructive analysis, 394 obvious?, 25-28 true for reals, 15

international date line, 104 inverse function theorem

alternative approach, 383-386 gives implicit function theorem, 323 many-dimensional, 318 one-dimensional, 102, 378

inverses in C(U, U), 317, 566 irrationality of

e, 94 7?, 441 7T, 449 v/2, 406

irrelevant m, 255 isolated points, 250

Jacobian determinant, 381 matrix, 122

Jensen's inequality, 424, 474

Kant, 28, 344 kindness to animals, 490 Krein-Milman for Mn, 421 Kronecker's lemma, 440

Lagrangian limitations, 334 method, 331-332 necessity, 330 sufficiency, 333

Leader, examples, 505 left and right derivative, 398 Legendre polynomials, 523-524, 540 Leibniz rule, 560 length of curve, 213-220, 497 L'Hopital's rule, 138, 430 limits

general view of, 371-376 in metric spaces, 231 in normed spaces, 231 more general than sequences, 54-56 pointwise, 265 sequences in Rm, 46 sequences in ordered fields, 4-7 uniform, 265

limsup and liminf, 39 lion hunting

in C, 42 in R, 15-16, 57, 468-469 in Mm, 47-48

Liouville, 409, 452

Lipschitz condition, 291 equivalence, 235

logarithm for (0,oo), 100-102, 453, 473 non-existence for C \ {0}, 104-105, 298-

300 what preceded, 450

Markov chains, 551-555 maxima and minima, 57, 147-153, 184-192,

328-334 Maxwell

hill and dale theorem, 157 prefers coordinate free methods, 46, 117

mean value inequality for complex differentiation, 274 for reals, 18-20, 22, 35, 59 many-dimensional, 131-132

mean value theorem Cauchy's, 430 discussion of, 59 fails in higher dimensions, 133 for higher derivatives, 428 for integrals, 467 statement and proof, 60

metric as measure of similarity, 264 British railway non-stop, 231 British railway stopping, 231 complete, 249 completion, 343 definition, 230 derived from norm, 230 discrete, 258 Hausdorff, 513-514 Lipschitz equivalent, 235 totally bounded, 259

Mobius transformation, 242-248, 450 monotone convergence

for some integrals, 528 for sums, 444

neighbourhood, 50, 233 non-Euclidean geometry, 344-345 norm

all equivalent on W1, 235 completion, 338, 343 definition, 229 Euclidean, 44 operator, 124, 240, 457, 512 sup, 261 uniform, 261, 263

notation, see also spaces Dijg, 144 Dj9, 122 g,ij, 141

Index 589

9,3 > 1 2 2

t] 568 x • y and (x ,y) , 43 z*, 114 || || and || • ||, 229 non-uniform, 396-397

nowhere different iable continuous function, 531

one-one, 17 one-to-one, 17 onto, 17 open problems, 78, 441 open sets

can be closed, 259 complement of closed sets, 50, 233 definition for metric space, 232 definition for Mm, 49 key properties, 51, 233

operator norm, 124, 240, 457, 512 ordered fields, 3, 357-360, 387-389 orthogonal polynomials, 522

parallelogram law, 574-576 partial derivatives

and Jacobian matrix, 122 and possible differentiability, 140, 154 definition, 122 notation, 122, 141, 144, 377-380, 397 symmetry of second, 143, 156

partition, see dissection pass the parcel, 320 piecewise definitions, 399 placeholder, 229, 331, 396 pointwise compared with uniform, 64-65, 265,

268 power series

addition, 432 and differential equations, 278, 544 composition, 444 convergence, 70 differentiation, 276, 537-539 limitations, 137, 282 many variable, 443 multiplication, 91 on circle of convergence, 70, 78 real, 277 uniform convergence, 275 uniqueness, 277 zeros isolated, 543

powers beat polynomials, 408, 451 definition of, 105-109, 278-280, 535-537

primary schools, Hungarian, 362 primes, infinitely many, Euler's proof, 447 probability theory, 210-213, 228 Ptolomey's inequality, 416

quantum mechanics, 28

radical reconstructions of analysis, 354, 391-394

radius of convergence, see also power series, 70, 76, 275, 433

rationals countable, 363 dense in reals, 12 not good for analysis, 1-3

reals, see also continuum, models for and fundamental axiom, 9 existence, 348-353 uncountable, 17, 363, 419 uniqueness, 359-360

reduction of order, 562 restriction of a function, 8 Riemann integral, see integral Riemann-Lebesgue lemma, 547 Rolle's theorem

examination of proof, 426 interesting use, 63-64 statement and proof, 60-62

Routh's rule, 461 routine, 50 Russell's paradox, 354

saddle, 150 sandwich lemma, 7 Schur complement, 461 Schwarz, area counterexample, 218 sequential compactness, 258 Shannon's theorem, 224-228 Simpson's rule, 473 singular points, see critical points slide rule, 107 solution of linear equations via

Gauss-Seidel method, 570 Jacobi method, 570

sovereigns, golden, 79-80 space-filling curve, 532 spaces

C([a, 6]), || | | i ) , 252, 264 C([a, 6]), || | | 2 ) , 252, 264 C([a, 6]), || Hoc), 264 C([a, 6]), || | | p ) , 510 co, 256 Z1, 253 I2, 508 /°°, 256 IP, 511 c(utU), 566 C(U,V), 240, 512 5oo, 251

spectral radius, 567-569 squeeze lemma, 7

590 Index

Stirling's formula, simple versions, 199, 226, 480

successive approximation, 311-313 successive bisection, see lion hunting summation methods, 434-437 sums, see also power series, Fourier series,

term by term and convergence tests absolute convergence, 68 conditionally convergent, 75 convergence, 67 dominated convergence, 82 equivalent to sequences, 67, 272 Fubini's theorem, 87 monotone convergence, 444 rearranged, 79, 83, 441

sup norm, 261 supremum

and fundamental axiom, 37 definition, 32 existence, 33 use, 34-37

surjective function, 17, 449 symmetric

linear map, 457 matrix, diagonalisable, 419

taxicab argument, 140 Taylor series, see power series Taylor theorems

best for examination, 180 Cauchy's counterexample, 137 depend on fundamental axiom, 138 global in R, 136, 180, 428 in R, 135-139 little practical use, 181, 281 local in R, 136 local i n R n , 144-145, 147 strong counterexample, 538

Tchebychev inequality, 211 polynomials, 427-429 spelling, 428

term by term differentiation, 272, 276, 286 integration, 272

Thor's drinking horn, 498 tin cans, 331 Torricelli's trumpet, 498 total boundedness, 259 total variation, 493 transcendentals, existence of

Cantor's proof, 363 Liouville's proof, 409

Trapezium rule, 472 trigonometric functions, 95-99, 137, 301, 496-

497 troublesome operations, 290

uniform continuity, 64-66, 173, 261 convergence, 265-273 norm, 261

uniqueness antiderivative, 20, 177 completions, 336-338 decimal expansion, 13 Fourier series, 283 limit, 4, 46, 231 power series, 277 reals, 359-360 solution of differential equations, 289-292

universal chord theorem, 414

variation of parameters, 563 Vieta's formula for 7r, 448 Vitali's paradox, 163 volume of an n-dimensional sphere, 221

Wallis formula for 7r, 446 integrals of powers, 471

Weierstrass M-test, 273 non-existence of minima, 189-192, 516-

518 polynomial approximation, 520

well ordering of integers, 10, 31 witch's hat

ordinary, 266 tall, 268

Wronskian, 303-304, 561-562

X, marks the spot, 590

young lady, deep, 362 young man, deep, 309, 505

Zeno, 25-29 zeros

interlaced, 525, 561 isolated, 543

zeta function, brief appearance, 282

Titles in This Series

62 T. W . Korner, A companion to analysis: A second first and first second course in

analysis, 2004

61 Thomas A. Ivey and J. M. Landsberg, Cart an for beginners: Differential geometry via

moving frames and exterior differential systems, 2003

60 A lberto Candel and Lawrence Conlon, Foliations II, 2003

59 Steven H. Weintraub, Representation theory of finite groups: algebra and arithmetic,

2003

58 Cedric Villani, Topics in optimal transportation, 2003

57 Robert P lato , Concise numerical mathematics, 2003

56 E. B . Vinberg, A course in algebra, 2003

55 C. Herbert Clemens, A scrapbook of complex curve theory, second edition, 2003

54 Alexander Barvinok, A course in convexity, 2002

53 Henryk Iwaniec, Spectral methods of automorphic forms, 2002

52 Ilka Agricola and Thomas Friedrich, Global analysis: Differential forms in analysis,

geometry and physics, 2002

51 Y. A. Abramovich and C. D . Aliprantis, Problems in operator theory, 2002

50 Y. A. Abramovich and C. D . Aliprantis, An invitation to operator theory, 2002

49 John R. Harper, Secondary cohomology operations, 2002

48 Y. Eliashberg and N . Mishachev, Introduction to the ^-principle, 2002 47 A. Yu. Kitaev, A. H. Shen, and M. N . Vyalyi , Classical and quantum computation,

2002 46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and

Lie groups, 2002

45 Inder K. Rana, An introduction to measure and integration, second edition, 2002

44 J im Agler and John E. M c C a r t h y , Pick interpolation and Hilbert function spaces, 2002

43 N . V. Krylov, Introduction to the theory of random processes, 2002

42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002

41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002

40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable,

2002

39 Larry C. Grove, Classical groups and geometric algebra, 2002

38 Elton P. Hsu, Stochastic analysis on manifolds, 2002

37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular

group, 2001

36 Martin Schechter, Principles of functional analysis, second edition, 2002

35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001

34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001

33 Dmitr i Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001

32 Robert G. Bartle , A modern theory of integration, 2001

31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods

of financial mathematics, 2001

30 J. C. McConnel l and J. C. Robson, Noncommutative Noetherian rings, 2001

29 Javier Duoandikoetxea , Fourier analysis, 2001

28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000

27 Thierry Aubin, A course in differential geometry, 2001

26 Rolf Berndt , An introduction to symplectic geometry, 2001

25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000

TITLES IN THIS SERIES

24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000

23 A lberto Candel and Lawrence Conlon, Foliations I, 2000

22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov

dimension, 2000

21 John B. Conway, A course in operator theory, 2000

20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999

19 Lawrence C. Evans, Partial differential equations, 1998

18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic

tools for every mathematician, 1997

17 Henryk Iwaniec, Topics in classical automorphic forms, 1997

16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator

algebras. Volume II: Advanced theory, 1997

15 Richard V . Kadison and John R. Ringrose, Fundamentals of the theory of operator

algebras. Volume I: Elementary theory, 1997

14 Elliott H. Lieb and Michael Loss, Analysis, 1997

13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996

12 N . V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996

11 Jacques Dixmier , Enveloping algebras, 1996 Printing

10 Barry Simon, Representations of finite and compact groups, 1996

9 D ino Lorenzini, An invitation to arithmetic geometry, 1996

8 Winfried Just and Martin Weese , Discovering modern set theory. I: The basics, 1996

7 Gerald J. Janusz, Algebraic number fields, second edition, 1996

6 Jens Carsten Jantzen, Lectures on quantum groups, 1996

5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995

4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 Wil l iam W . A d a m s and Phi l ippe Loustaunau, An introduction to Grobner bases,

1994 2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity,

1993 1 Ethan Akin, The general topology of dynamical systems, 1993