Problem Books in Mathematics

Edited by P.R. Halmos

Problem Books in Mathematics

Series Editor: P.R. Halmos

Polynomials by Edward J. Barbeau

Problems in Geometry by Marcel Berger, Pierre Pansu, Jean-Pic Berry, and Xavier Saint-Raymond

Problem Book for First Year Calculus by George W. B/uman

Exercises in Probability by T. Cacoullos

An Introduction to Hilbert Space and Quantum Logic by David W. Cohen

Unsolved Problems in Geometry by Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy

Problems in Analysis by Bernard R. Gelballm

Problems in Real and Complex Analysis by Bernard R. Gelballm

Theorems and Counterexamples in Mathematics by Bernard R. Gelbaum and Jolm M.H. Olmsted

Exercises in Integration by Claude George

Algebraic Logic by S.G. Gindikin

Unsolved Problems in Number Theory by Richard K. Guy

An Outline of Set Theory by James M. Henle

(continued after index)

Bernard R. Gelbaum John M.R. Olmsted

Theorems and Counterexamples in Mathematics

With 24 Illustrations

Springer Science+Business Media, LLC

Bernard R. Gelbaum Department of Mathematics State University of New York at Buffalo Buffalo, New York 14214-3093 USA

John M.H. Olmsted Department of Mathematics Southern Illinois University Carbondale, Illinois 62901 USA

Editor

Paul R. Halmos Department of Mathematics Santa Clara University Santa Clara, California 95053, USA

Mathematical Subject Classifications: 00A07

Library of Congress Cataloging-in-Publication Data Gelbaum, 8ernard R.

Theorems and counterexamples in mathematics I 8ernard R. Gelbaum, lohn M.H. Olmsted.

p. cm - (Problem books in mathematics) Includes bibliographical references and index. 1. Mathematics. 1. Olmsted, lohn Meigs Hubbell, 1911-

II. TitIe. III. Series. QA36.G45 1990 51O-dc20 90-9899

Printed on acid-free paper

© 1990 Springer Science+Business Media New York Originally published by Springer-Verlag New York Inc. in 1990 Softcover reprint of the hardcover 1 st edition 1990

CIP

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC except for brief excerpts in connection with reviews or scho1arly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereaf­ter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Photocomposed copy prepared by the authors using T EX,

9 8 7 6 5 4 3 2 (Second corrected printing)

ISBN 978-1-4612-6975-5 ISBN 978-1-4612-0993-5 (eBook) DOI 10.1007/978-1-4612-0993-5

PREFACE

The gratifying response to Counterexamples in analysis (CEA) was followed, when the book went out of print, by expressions of dismay from those who were unable to acquire it.

The connection of the present volume with CEA is clear, although the sights here are set higher. In the quarter-century since the appearance of CEA, mathematical education has taken some large steps reflected in both the undergraduate and graduate curricula. What was once taken as very new, remote, or arcane is now a well-established part of mathematical study and discourse. Consequently the approach here is designed to match the observed progress. The contents are intended to provide graduate and ad­vanced undergraduate students as well as the general mathematical public with a modern treatment of some theorems and examples that constitute a rounding out and elaboration of the standard parts of algebra, analysis, geometry, logic, probability, set theory, and topology.

The items included are presented in the spirit of a conversation among mathematicians who know the language but are interested in some of the ramifications of the subjects with which they routinely deal. Although such an approach might be construed as demanding, there is an extensive GLOSSARY jlNDEX where all but the most familiar notions are clearly defined and explained. The object ofthe body of the text is more to enhance what the reader already knows than to review definitions and notations that have become part of every mathematician's working context.

Thus terms such as complete metric space, O'-ring, Hamel basis, linear programming, {logical] consistency, undecidability, Cauchy net, stochastic independence, etc. are often used without further comment, in which case they are italicized to indicate that they are carefully defined and explained in the GLOSSARY jINDEX.

The presentation of the material in the book follows the pattern below:

A definition is provided either in the text proper or in the GLOS­SARY jINDEX. The term or concept defined is usually italicized at some point in the text.

ii. A THEOREM for which proofs can be found in most textbooks and monographs is stated often without proof and always with at least one reference.

iii A result that has not yet been expounded in a textbook or monograph is given with at least one reference and, as space permits, with a proof, an outline of a proof, or with no proof at all.

w Validation of a counterexample is provided in one of three ways: a. As an Exercise (with a Hint if more than a routine calculation is

involved). b. As an Example and, as space permits, with a proof, an outline

v

vi Preface

of a proof, or with no proof at all. Wherever full details are not given at least one reference is provided.

c. As a simple statement and/or description together with at least one reference.

Preceding the contents there is a GUIDE to the principal items treated. We hope this book will offer at least as much information and pleasure

as CEA seems to have done to (the previous generation of) its readers. The current printing incorporates corrections, many brought to our

attention by RB. Burckel, G. Myerson, and C. Wells, to whom we offer our thanks.

State University of New York at Buffalo B. R G.

Carbondale, Illinois J.M.H.O.

Contents

Preface v

Guide ix

1 Algebra 1.1 Group Theory

1.1.1 Axioms 1 1.1.2 Subgroups 2 1.1.3 Exact versus splitting sequences 4 1.1.4 The functional equation: f(x + y) = f(x) + fey) 5 1.1.5 Free groups; free topological groups 9 1.1.6 Finite simple groups 18

1.2 Algebras 1.2.1 Division algebras ("noncommutative fields") 19 1.2.2 General algebras 20 1.2.3 Miscellany 22

1.3 Linear Algebra 1.3.1 Finite-dimensional vector spaces 25 1.3.2 General vector spaces 31 1.3.3 Linear programming 37

2 Analysis 2.1 Classical Real Analysis

2.1.1 RX 42 2.1.2 Derivatives and extrema 2.1.3 Convergence of sequences and series 2.1.4 RXxY

2.2 Measure Theory 2.2.1 Measurable and nonmeasurable sets 2.2.2 Measurable and nonmeasurable functions 2.2.3 Group-invariant measures

2.3 Topological Vector Spaces 2.3.1 Bases 2.3.2 Dual spaces and reflexivity 2.3.3 Special subsets of Banach spaces 2.3.4 Function spaces

2.4 Topological Algebras 2.4.1 Derivations 2.4.2 Semisimplicity

2.5 Differential Equations 2.5.1 VVronskians 2.5.2 Existence/uniqueness theorems

53 66 95

103 132 143

156 162 165 168

172 174

177 177

vii

viii

2.6 Complex Variable Theory 2.6.1 Morera's theorem 2.6.2 Natural boundaries 2.6.3 Square roots 2.6.4 Uniform approximation 2.6.5 Rouche's theorem 2.6.6 Bieberbach's conjecture

3 Geometry/Topology 3.1 Euclidean Geometry

3.1.1 Axioms of Euclidean geometry 3.1.2 Topology of the Euclidean plane

3.2 Topological Spaces 3.2.1 Metric spaces 3.2.2 General topological spaces

3.3 Exotica in Differential Topology

4 Probability Theory 4.1 Independence 4.2 Stochastic Processes 4.3 Transition Matrices

5 Foundations 5.1 Logic 5.2 Set Theory

Bibliography

Supplemental Bibliography

Symbol List

Glossary /Index

Contents

180 180 183 183 184 184

186 190

198 200 208

210 216 221

223 229

233

243

249

257

GUIDE

The list below provides the sequence in which the essential items in the book are presented.

In this GUIDE and in the text proper, the boldface numbers a.b.c.d. e following an [Item] indicate [Item] d on page e in Chapter a, Section b, Subsection c; similarly boldface numbers a.b.c. d following an [Item] indicate [Item] c on page d in Chapter a, Section b; e.g., Example 1.3.2.7. 35. refers to the seventh Example on page 35 in Subsection 2 of Section 3 of Chapter 1; LEMMA 4.2.1. 218. refers to the first LEMMA on page 218 in Section 2 in Chapter 4.

Group Theory

1. Faulty group axioms. Example 1.1.1.1. 2, Remark 1.1.1.1. 2.

2. Lagrange's theorem and the failure of its converse. THEOREM 1.1.2.1 3, Exercise 1.1.2.1. 3.

3. Cosets as equivalence classes. Exercise 1.1.2.2. 3.

4. A symmetric and transitive relation need not be reflexive. Exercise 1.1.2.3. 3.

5. A subgroup H of a group G is normal iff every left (right) coset of H is a right (left) coset of H.

Exercise 1.1.2.4. 3 6. If G : H is the smallest prime divisor p of #(G) then H is a normal

subgroup. THEOREM 1.1.2.2. 4.

7. An exact sequence that fails to split. Example 1.1.3.1. 5.

8. If the topological group H contains a countable dense set and if the ho­momorphism h : G 1--+ H of the locally compact group G is measurable on some set P of positive measure then h is continuous (everywhere).

THEOREM 1.1.4.1. 5. 9. If A is a set of positive (Haar) measure in a locally compact group then

AA -1 contains a neighborhood of the identity. pages 5-6.

10. The existence of a Hamel basis for lR over Q implies the existence in lR of a set that is not Lebesgue measurable.

page 6.

ix

x Guide

11. If f (in 1R1R) is a nonmeasurable function that is a solution of the func­tional equation f(x + y) = f(x) + f(y) then a) f is unbounded both above and below in every nonempty open interval and b) if R is one of

the relations <,:5, >,;::: and ER,Ol ~f {x : f(x) R a}, then for all a in R and for every open set U, ER,Ol n U is dense in U.

Exercise 1.1.4.1. 6. 12. There are nonmeasurable midpoint-convex functions.

Exercise 1.1.4.2. 7. 13. There exists a Hamel basis B for IR over Q and A(B) = O.

THEOREM 1.1.4.2. 7. 14. For the Cantor set Co: Co + Co = [0,2].

Exercise 1.1.4.3. 7, Note 1.1.4.1. 7. 15. The Cantor set Co contains a Hamel basis for IR over Q.

Exercise 1.1.4.4. 7. 16. Finiteness is a Quotient Lifting (QL) property of groups.

Example 1.1.4.1. 8. 17. Abelianity is not a QL property of groups.

Example 1.1.4.2. 8. 18. Solvability is a QL property of groups.

Exercise 1.1.4.5. 8. 19. Compactness is a QL property of locally compact topological groups.

Example 1.1.4.3. 9. 20. If X is a set there is a free group on X.

Exercise 1.1.5.1. 9. 21. The free group on X.

Note 1.1.5.1. 10. 22. Every group G is the quotient group of some free group F(X).

Exercise 1.1.5.2. 10. 23. A group G can be the quotient group of different free groups.

Note 1.1.5.2. 11. 24. The undecidability of the word problem for groups.

Note 1.1.5.2. 11. 25. There is a finitely presented group containing a finitely generated sub­

group for which there is no finite presentation. Note 1.1.5.2. 11.

26. An infinite group G presented by a finite set {Xl, ... , xn } of generators and a finite set

of identities. Note 1.1.5.2. 11.

27. The Morse-Hedlund nonnilpotent semigroup ~ generated by three nil­potent elements.

pages 11-12.

Guide xi

28. Every quaternion q is a square. Exercise 1.1.5.3. 13.

29. Two pure quaternions commute iff they are linearly dependent over lR. Exercise 1.1.5.4. 13.

30. If X is a completely regular topological space there is a free topological group Ftop(X) on X.

THEOREM 1.1.5.1. 14. 31. A quaternion q is of norm 1: Iql = 1 iff q is a commutator.

THEOREM 1.1.5.2. 15. 32. The commutator subgroup of 1Hl* is the set of quaternions of norm 1:

Q (1Hl*) = {q : q E 1Hl, Iql = 1 }. Note 1.1.5.3. 15.

33. In 1Hl* there is a free subset T such that #(T) = # (lR). Remark 1.1.5.1. 17.

34. A faulty commutative diagram. Example 1.1.5.1. 18.

35. The square root function is not continuous on T. Exercise 1.1.5.5. 18.

36. The classification of finite simple groups. Subsection 1.1.6. 18.

37. For two (different) primes p and q, are the natural numbers

relatively prime?

Algebras

pq - 1 qP-1 --and--p-1 q-1

Note 1.1.6.1. 19.

38. Over 1Hl, a polynomial of degree two and for which there are infinitely many zeros.

Example 1.2.1.1. 19.

39. There are infinitely many different quaternions of the form qiq-l. Exercise 1.2.1.1. 20.

40. If the quaternion r is such that r2 + 1 = 0 then for some quaternion def • -1 q, r = rq = qlq

THEOREM 1.2.1.1. 20. 41. A nonassociative algebra.

Exercise 1.2.2.1. 21. 42. The Jacobi identity.

Exercise 1.2.2.2. 21. 43. Lie algebras and groups of Lie type.

Remark 1.2.2.1. 21.

xii Guide

44. The Cayley algebra. Exercise 1.2.2.3. 22.

45. Milnor's classification of the alternative division algebras. page 22.

46. e cannot be ordered. Exercise 1.2.3.1. 22.

47. A field with two different orders. Exercise 1.2.3.2. 23.

48. Q is not complete. Exercise 1.2.3.3. 23.

49. A non-Archimedeanly ordered field. Exercise 1.2.3.4. 23.

50. Two complete Archimedeanly ordered fields are order-isomorphic. Note 1.2.3.1. 23.

51. An ordered field lK that is not embeddable in lR so that the orders in lR and in lK are consistent.

Exercise 1.2.3.5. 23. 52. A complete Archimedeanly ordered field is Cauchy complete.

Exercise 1.2.3.6. 24. 53. A characterization of Cauchy nets in lR.

Exercise 1.2.3.7. 25. 54. A field that is Cauchy complete and not complete.

Example 1.2.3.1. 25.

Linear Algebra

55. The set [V]sing of singular endomorphisms of an n-dimensional vector 2

space V over e is a closed nowhere dense null set in en . THEOREM 1.3.1.1. 26.

def 2 56. The set [V] \ [V]sing = [V]inv is a dense (open) subset of en .

COROLLARY 1.3.1.1. 26. 2

57. In en the set V of diagonable n x n matrices is nowhere dense; its complement is open and dense; An2 (V) = o.

Exercise 1.3.1.1. 26. 58. A pair of commuting nondiagonable matrices.

Exercise 1.3.1.2. 27. 59. A pair of commuting matrices that are not simultaneously "Jordaniz­

able." Exercise 1.3.1.2. 27.

60. If a finite-dimensional vector space over lR is the finite union of sub­spaces, one of those subspaces is the whole space.

THEOREM 1.3.1.2. 27, Remark 1.3.1.1. 28.

Guide xiii

61. A vector space that is the union of three proper subspaces. Exercise 1.3.1.3. 28.

62. The Moore-Penrose inverse. Exercise 1.3.1.4. 28.

63. A failure of the GauB-Seidel algorithm. Example 1.3.1.1. 29.

64. The failure for vector space homomorphisms of: (ST = J) => (T S = J). Example 1.3.2.1. 31.

65. A vector space endomorphism without eigenvalues. Example 1.3.2.2. 32.

66. A vector space endomorphism for which the spectrum is C \ {O}. Example 1.3.2.3. 32.

67. A vector space endomorphism for which the spectrum is empty. Example 1.3.2.4. 32.

68. A vector space endomorphism for which the spectrum is C. Example 1.3.2.5. 33.

69. A Banach space containing a dense proper subspace; discontinuous endomorphislll8; absence of non-Hamel bases; for a Banach space V, T* exists in [V*] implies T is continuous.

page 34. 70. A Euclidean vector space endomorphism having no adjoint.

Example 1.3.2.6. 34. 71. A noninvertible Euclidean space endomorphism that is an isometry.

Example 1.3.2.7. 35. 72. Sylvester's Law of Inertia.

THEOREM 1.3.2.1. 35. 73. The set of continuous invertible endomorphisms of Hilbert space is

connected. THEOREM 1.3.2.2. 36.

74. A commutative Banach algebra in which the set of invertible elements is not connected.

Example 1.3.2.8. 37. 75. There is no polynomial bound on the number of steps required to

complete the simplex algorithm in linear programming. page 38.

76. The number of steps required to complete GauBian elimination is poly­nomially bounded.

Example 1.3.3.1. 38. 77. Karmarkar's linear programming algorithm for which the number of

steps required for completion is polynomially bounded. page 38.

78. A linear programming problem for which the simplex algorithm cycles. Example 1.3.3.2. 39.

xiv Guide

79. The Bland and Charnes algorithms. pages 40-41.

Classical Real Analysis

80. The set Cont(J) is a G6. THEOREM 2.1.1.1. 43.

81. The set Discont(J) is an Fu' Exercise 2.1.1.1. 43.

82. An Fu that is not closed. Example 2.1.1.1. 43.

83. Baire's category theorem and corollaries. THEOREM 2.1.1.2. 43, COROLLARY 2.1.1.1. 43,

COROLLARY 2.1.1.2. 44. 84. A modified version of Baire's category theorem.

Exercise 2.1.1.2. 44. 85. In 1R a sequence of dense sets having nonempty interiors and for which

the intersection is not dense. Example 2.1.1.2. 44.

86. If f is the limit of continuous functions on a complete metric space X then Cont(J) is dense in X.

THEOREM 2.1.1.3. 45, Remark 2.1.1.1. 45, Exercise 2.1.1.4. 45.

87. If F is closed and FO = 0 then F is nowhere dense. Exercise 2.1.1.3. 45.

88. A nowhere continuous function !1 such that 1ft I is constant; a non­measurable function 12 such that 1121 is constant.

Exercise 2.1.1.5. 47. 89. A somewhere continuous function not the limit of continuous func­

tions; a nonmeasurable function somewhere continuous; a discontinu­ous function continuous almost everywhere; a discontinuous function equal almost everywhere to a continuous function; a nonmeasurable function that is somewhere differentiable.

Exercise 2.1.1.6. 47. 90. A continuous locally bounded but unbounded function on a bounded

set. Exercise 2.1.1.7. 47.

91. A continuous function having neither a maximum nor a mInlmUm value; a bijective bicontinuous function mapping a bounded set onto an unbounded set.

Exercise 2.1.1.8. 47.

Guide xv

92. A bounded function defined on a compact set and having neither a maximum nor a minimum value there.

Exercise 2.1.1.9. 48. 93. A nowhere semicontinuous function f defined on a compact set and

such that liminf f(x) == -1 < f(x) < 1 = limsupf(x) == 1. Exercise 2.1.1.10. 48.

94. A nonconstant continuous periodic function in ~IR has a least positive period.

THEOREM 2.1.1.4. 48. 95. A nonconstant periodic function without a smallest positive period.

Exercise 2.1.1.11. 48. 96. For A an arbitrary Fu in ~, a function f such that Discont(f) = A.

Exercise 2.1.1.12. 48. 97. If f in ~IR is monotone then # (Discont(f)) ::; # (N); a function for f

which Discont(f) = Q. Exercise 2.1.1.13. 49.

98. For a positive sequence {dn}nEN such that E:=l dn < 00 and a se-

quence S ~f {an}nEN contained in~, a monotone function f such that Discont(f) = Sand

Exercise 2.1.1.14. 49. 99. A continuous nowhere monotone and nowhere differentiable function.

Exercise 2.1.1.15. 50. 100. A function H : [0, 1]1--+ ~ that is zero a.e. and maps every nonempty

subinterval (a, b) onto R Example 2.1.1.3. 51.

101. Properties of k-ary representations. Exercise 2.1.1.16. 52.

102. Two maps f and 8 such that f 0 8 is the identity and 8 0 f is not the identity.

Exercise 2.1.1.17. 52. 103. Every point of the Cantor set Co is a point of condensation.

Exercise 2.1.1.18. 52. 104. A differentiable function with a discontinuous derivative; a differen­

tiable function with an unbounded derivative; a differentiable function with a bounded derivative that has neither a maximum nor a minimum value.

Exercise 2.1.2.1. 53. 105. A derivative cannot be discontinuous everywhere.

Remark 2.1.2.1. 53.

xvi Guide

106. If a sequence of derivatives converges uniformly on a compact interval I and if the sequence of corresponding functions converges at some point of I then the sequence of functions converges uniformly on I.

THEOREM 2.1.2.1. 53. 107. A sequence of functions for which the sequence of derivatives converges

uniformly although the sequence of functions diverges everywhere. Note 2.1.2.1. 54.

108. If a function h defined on a compact interval I is of bounded varia­tion on I and also enjoys the intermediate value property then h is continuous.

THEOREM 2.1.2.2. 54. 109. If a derivative I' is of bounded variation on a compact interval I then

l' is continuous. COROLLARY 2.1.2.1. 54.

110. Inclusion and noninclusion relations among the sets BV(I), BV (JR), AC(I), and AC (JR).

Remark 2.1.2.3. 55. 111. On [0,1], a strictly increasing function for which the derivative is zero

almost everywhere. Example 2.1.2.1. 55.

112. A characterization of null sets in lR. Exercise 2.1.2.2. 56.

113. A set A in JR is a null set iff A is a subset of the set where some monotone function fails to be differentiable.

THEOREM 2.1.2.3. 56. 114. For a given sequence S in ]R a monotone function 1 such that

Discont(J) = Nondiff(J) = S.

Exercise 2.1.2.3. 57. 115. A differentiable function monotone in no interval adjoining one of the

points where the function achieves its minimum value. Exercise 2.1.2.4. 57.

116. A function for which the set of sites of local maxima is dense and for which the set of sites of local minima is also dense.

Example 2.1.2.2. 58.

117. If h E ]RIR, if h is continuous, and if h has precisely one site of a local maximum resp. minimum and is unbounded above resp. below then h has at least one site of a local minimum resp. maximum.

Exercise 2.1.2.5. 60. 118. Functions, each with precisely one site of an extremum, and unbounded

both above and below. Example 2.1.2.3. 60.

119. A nonmeasurable function that is infinitely differentiable at some point. Remark 2.1.2.5. 61.

Guide xvii

120. An infinitely differentiable function for which the corresponding Mac­laurin series represents the function at just one point.

Example 2.1.2.4. 61. 121. Bridging functions.

Exercises 2.1.2.6. 62, 2.1.2.7. 62, 2.1.2.8 62, 2.1.2.9 63. 122. A differentiable function for which the derivative is not Lebesgue inte­

grable. Example 2.1.2.5. 63.

123. A uniformly bounded sequence of Riemann integrable functions con­verging everywhere to a function that is not Riemann integrable on any nonempty open interval.

Exercise 2.1.2.10. 64. 124. A Riemann integrable function having no primitive.

Exercises 2.1.2.11. 64, 2.1.2.12. 65. 125. A function with a derivative that is not Riemann integrable.

Exercise 2.1.2.13. 65. 126. An indefinite integral that is differentiable everywhere but is not a

primitive of the integrand. Exercise 2.1.2.14. 65.

127. A minimal set of criteria for absolute continuity. Exercise 2.1.2.15. 65, Example 2.1.2.6. 65.

128. Relationships between bounded variation and continuity. Exercise 2.1.2.16 65, Example 2.1.2.7. 66.

129. The composition of two absolutely continuous functions can fail to be absolutely continuous.

Exercise 2.1.2.17. 66, Example 2.1.2.8. 66. 130. For a given closed set A in lR a sequence {an }nEN for which the set of

limit points is A. Exercise 2.1.3.1. 67.

131. A divergent series such that for each p in N, the sequence {sn }nEN of partial sums satisfies: limn ...... oo ISn+p - snl = O.

Exercise 2.1.3.2. 67. 132. For a strictly increasing sequence {v(n)}nEN in N, a divergent sequence

{an}nEN such that limn ...... oo lav(n) - ani = o. Exercise 2.1.3.3. 67.

133. For a sequence {v(n)}nEN in N and such that v(n) -+ 00 as n -+ 00, a divergent unbounded sequence {an}nEN such that

lim lav(n) - ani = o. n ...... oo

Exercise 2.1.3.4. 67. 134. Strict inequalities for the functionals lim sup, lim inf.

Exercise 2.1.3.5. 67.

xviii Guide

135. Identities for the set functions lim sup, lim inf. Exercise 2.1.3.5. 68.

136. In JR a decreasing sequence {An}nEN of sets such that a) for all n, # (An) = # (JR) and b) nnEN An = 0.

Exercise 2.1.3.6. 68. 137. Criteria for absolute convergence of numerical series.

Exercise 2.1.3.7. 69. 138. The Riemann derangement theorem.

Exercise 2.1.3.8. 69. 139. The Steinitz derangement theorem.

THEOREM 2.1.3.1. 70. 140. The Sierpinski derangement theorem.

THEOREM 2.1.3.2. 70. 141. Another derangement theorem of Sierpinski.

Remark 2.1.3.3. 70. 142. A special case of the Steinitz derangement theorem.

Exercise 2.1.3.9. 71. 143. Subseries of convergent and divergent numerical series.

Exercise 2.1.3.10. 71. 144. A divergent series E:'l an for which limn -+oo an = o.

Exercise 2.1.3.11. 72. 145. A convergent series that dominates a divergent series.

Exercise 2.1.3.12. 72. 146. A convergent series that absolutely dominates a divergent series.

Exercise 2.1.3.13. 72. 147. The absence of a universal comparison sequence of positive series.

THEOREM 2.1.3.3. 72. 148. A divergent series summable (C,l).

Example 2.1.3.1. 74. 149. Fejer's kernel.

Exercise 2.1.3.14. 74. 150. Fejer's theorem.

Exercise 2.1.3.15. 75. 151. Two Toeplitz matrices.

Exercises 2.1.3.16. 76, 2.1.3.17. 77. 152. Partial ordering among summability methods.

page 76. 153. Absence of a universal sequence of Toeplitz matrices.

THEOREM 2.1.3.4. 77. 154. Toeplitz matrices and Z f-+ eZ •

Exercise 2.1.3.18. 79. 155. Counterexamples to weakened versions of the alternating series theo­

rem. Exercise 2.1.3.19. 79.

Guide xix

156. Relations between rapidity of convergence to zero of the sequence of terms of a positive series and the convergence of the series ..

Exercise 2.1.3.20. 80, Remark 2.1.3.5. 80, Exercise 2.1.3.21. 80.

157. Failure of the ratio test, the generalized ratio test, the root test, and the generalized root test for convergence of positive series.

Exercises 2.1.3.22. 81, 2.1.3.23. 81, 2.1.3.24. 81. 158. Relations among the ratio and root tests.

Exercises 2.1.3.25. 82, 2.1.3.26. 82. 159. A divergent Cauchy product of convergent series.

Exercise 2.1.3.27. 82. 160. A convergent Cauchy product of divergent series.

Exercise 2.1.3.28. 82. 161. A Maclaurin series converging only at zero.

Exercise 2.1.3.29. 82. 162. For an arbitrary power series, a Coo function for which the given series

is the Maclaurin series. Example 2.1.3.2. 83, Remark 2.1.3.7. 84.

163. Convergence phenomena associated with power series. Example 2.1.3.3. 84.

164. Cantor's theorem about trigonometric series. THEOREM 2.1.3.5. 85, Note 2.1.3.2. 86.

165. A general form of Cantor's theorem. THEOREM 2.1.3.6. 86.

166. A faulty weakened general form of Cantor's theorem. Example 2.1.3.4. 86.

167. Abel's lemma. LEMMA 2.1.3.1. 87.

168. A trigonometric series that is not the Fourier series of a Lebesgue integrable function.

Examples 2.1.3.5. 87, 2.1.3.6. 87, Remark 2.1.3.9. 88. 169. A uniformly convergent Fourier series that is not dominated by a pos­

itive convergent series of' constants. Exercise 2.1.3.30. 88.

170. A continuous function vanishing at infinity and not the Fourier trans­form of a Lebesgue integrable function.

Example 2.1.3.7. 88. 171. The Fejer-Lebesgue and Kolmogorov examples of divergent Fourier se­

ries of integrable functions. page 89, Note 2.1.3.3. 89.

172. A continuous limit of a sequence of everywhere discontinuous functions. Exercise 2.1.3.31. 90.

173. A sequence {fn}nEN converging uniformly to zero and such that the sequence of derivatives diverges everywhere.

Exercise 2.1.3.32. 90.

xx Guide

174. An unbounded function that is the nonuniform limit of bounded func­tions.

Exercise 2.1.3.33. 90. 175. Discontinuous functions that are the nonuniform limits of continuous

functions. Exercises 2.1.3.34. 90, 2.1.3.35. 90, Remark 2.1.3.10. 91.

176. An instance in which the interchange of J and lim is valid although the limit is not uniform.

Exercise 2.1.3.36. 91. 177. A Riemann integrable limit of Riemann integrable functions where the

interchange of J and lim is not valid. Exercise 2.1.3.37. 92.

178. A function that is Lebesgue integrable, is not Riemann integrable, and is the nonuniform limit of uniformly bounded Riemann integrable func­tions.

Exercise 2.1.3.38. 92. 179. A power series in which the terms converge uniformly to zero and the

series does not converge uniformly. Exercise 2.1.3.39. 92.

180. A sequence {!n}nEN that converges nonuniformly to zero while the sequence {!2n}nEN converges uniformly (to zero).

Exercise 2.1.3.40. 93. 181. The failure of weakened versions of Dini's theorem.

Exercise 2.1.3.41. 93. 182. A sequence of functions converging uniformly to zero on [-I,IJ al­

though the sequence of their derivatives fails to converge on [-1, IJ. Exercise 2.1.3.42. 93.

183. A sequence converging uniformly on every proper subinterval of an interval and failing to converge uniformly on the interval.

Exercise 2.1.3.43. 93. 184. A sequence {!n}nEN converging uniformly to zero on [0,00) and such

that J[O,oo) !n(x) dx 1 00. Exercise 2.1.3.44. 93.

185. A power series that, for each continuous function !, converges uni­formly, via grouping of its terms, to !.

Example 2.1.3.8. 93, Note 2.1.3.4. 94. 186. A series of constants that, for each real number x, converges, via group­

ing of its terms, to x. Exercise 2.1.3.45. 94.

187. An instance of divergence of Newton's algorithm for locating the zeros of a function.

Example 2.1.3.9. 95. 188. Uniform convergence of nets.

Exercise 2.1.4.1. 95.

Guide xxi

189. A function I in ]R1R2 and continuous in each variable and not continuous in the pair.

Exercise 2.1.4.2. 95.

190. In ]R1R2 functions I discontinuous at (0,0) and continuous on certain curves through the origin.

Exercises 2.1.4.3. 96, 2.1.4.4. 96.

191. In ]R1R2 functions I nondifferentiable at (0,0) and having first partial derivatives everywhere.

Note 2.1.4.1. 96.

192. In ]R1R2 functions I for which exactly two of

lim lim I(x, y), lim lim I(x, y), and lim I(x, y) x-o y-o y_o x-a (x,y)-(O,O)

exist and are the equal. Exercise 2.1.4.5. 96.

193. In ]R1R2 functions I for which exactly one of

lim lim I(x, y), lim lim I(x, y), and lim I(x, y) x-Oy-O y-Ox-O (x,y)_(O,O)

exists. Exercise 2.1.4.6. 97.

194. The Moore-Osgood theorem. THEOREM 2.1.4.1. 97.

195. In ]R1R2 a function I for which both

lim lim I(x, y) and lim lim I(x, y) x-O y-O y-O x-O

exist but are not equal. Exercise 2.1.4.7. 97.

196. A false counterexample to the Moore-Osgood theorem. Exercise 2.1.4.8. 97.

197. In ]R1R2 a function I differentiable everywhere but for which Ix and Iy are discontinuous at (0,0).

Exercise 2.1.4.9. 98. 198. The law of the mean for functions of two variables.

page 98.

199. In ]R1R2 a function I such that Ix and Iy exist and are continuous but Ixy(O,O) ¥ lyx(O, 0).

Exercise 2.1.4.10. 98.

xxii Guide

200. In ]R1R2 a function f such that fy == 0 and yet f is not independent of y.

~xercise 2.1.4.11. 99, Note 2.1.4.2. 99.

201. In ]R1R2 a function f without local extrema, but with a local extremum at (0,0) on every line through (0,0).

Exercise 2.1.4.12. 99.

202. In ]R1R2 a function f such that

Exercise 2.1.4.13. 100.

203. In ]R1R2 a function f such that

1111 f(x,y)dxdy = 1

1111 f(x,y)dydx =-1.

Exercise 2.1.4.14. 100. 204. A double sequence in which repeated limits are unequal.

Exercise 2.1.4.15. 100. 205. Counterexamples to weakened versions of Fubini's theorem.

Note 2.1.4.3. 101. 206. Kolmogorov's solution of Hilbert's thirteenth problem.

Example 2.1.4.1. 101, THEOREM 2.1.4.2. 102.

Measure Theory

207. The essential equivalence of the procedures:

measure 1-+ nonnegative linear functional

nonnegative linear functional 1-+ measure.

Remark 2.2.1.1. 104. 208. A Hamel basis for ]R is measurable iff it is a null set.

THEOREM 2.2.1.1. 104. 209. No Hamel basis for ]R is Borel measurable.

THEOREM 2.2.1.2. 105. 210. A non~Borel subset of the Cantor set.

Remark 2.2.1.2. 105.

Guide xxiii

211. In every neighborhood of 0 in JR there is a Hamel basis for JR over Q. THEOREM 2.2.1.3. 105.

212. A nonmeasurable subset of JR. Example 2.2.1.1. 106.

213. In JR a subset M such that:

z. >.*(M) = 0 and >'*(M) = 00 (M is nonmeasurable)j zz. for any measurable set P: >.*(P n M) = 0 and >'*(P n M) = >.(P).

Example 2.2.1.2. 106 214. Every infinite subgroup of T is dense in Tj 1 x T is a nowhere dense

infinite subgroup of ']['2.

Exercise 2.2.1.1. 107. 215. A nowhere dense perfect set consisting entirely of transcendental num­

bers. Example 2.2.1.3. 108, Exercise 2.2.1.2. 108.

216. In [0,1] an Fer a) consisting entirely of transcendental numbers, b) of the first category, and c) of measure one.

Exercise 2.2.1.3. 109. 217. A null set H such that every point in JR is point of condensation of H.

Exercise 2.2.1.4. 109. 218. In some locally compact groups measurable subsets A and B such that

AB is not measurable. Examples 2.2.1.4. 109, 2.2.1.5. 110.

219. In JR a thick set of the first category. Example 2.2.1.6. 110.

220. Disjoint nowhere dense sets such that each point of each set is a limit point of the other set.

Exercise 2.2.1.5. 111. 221. Two countable ordinally dense sets are ordinally similar.

THEOREM 2.2.1.4. 111. 222. A nowhere dense set homeomorphic to a dense set.

Exercise 2.2.1.6. 112. 223. Dyadic spaces as pre-images of some compact sets.

LEMMA 2.2.1.1. 112. 224. A special kind of compact Hausdorff space.

Exercise 2.2.1.7 113. 225. A compact Hausdorff space that is not the continuous image of any

dyadic space. Exercise 2.2.1.8. 113.

226. The distinction between the length of an arc and the length of an arc­image.

Example 2.2.1.7. 114.

227. A nonrectifiable arc for which the arc-image is a line segment PQ. Example 2.2.1.7. 114.

xxiv Guide

228. A continuous map that carries a linear null set into a thick planar set. Example 2.2.1.8. 114.

229. A continuous map that carries a null set in R, into a nonmeasurable set (first example).

Example 2.2.1.8. 114. 230. For n greater than 1, in R,n nonrectifiable simple arc-images of positive

n-dimensional Lebesgue measure. Example 2.2.1.9. 115, Exercise 2.2.1.9. 117,

Note 2.2.1.3. 117. 231. In R,2 a Jordan curve-image of positive measure.

Examples 2.2.1.10. 117, 2.2.1.12. 123. 232. A compact convex set in a separable topological vector space is an

arc-image. Exercise 2.2.1.10. 117.

233. In R3 a set that, for given positive numbers." (arbitrarily small) and A (arbitrarily large), a) is homeomorphic to the unit ball of R3 and b) has a boundary for which the surface area is less than." but for which the three-dimensional Lebesgue measure is greater than A.

Example 2.2.1.11. 118, Exercise 2.2.1.11. 118, Remark 2.2.1.4. 121, Note 2.2.1.4. 121.

234. A faulty definition of surface area. Exercise 2.2.1.12. 123.

235. The Kakeya problem and a related problem. THEOREMS 2.2.1.5. 124, 2.2.1.6. 129.

236. When p = 3 the bisection-expansion procedure yields the optimal over­lap in the construction of the Perron tree.

Exercise 2.2.1.13. 129. 237. In R2 a nonmeasurable set meeting each line in at most two points.

Example 2.2.1.13. 130.

238. In RIR a function having a nonmeasurable graph. Exercise 2.2.1.14. 131.

239. In R2 regions without content. Examples 2.2.1.14. 131, 2.2.1.15. 131,

2.2.1.16. 131, Exercise 2.2.1.15. 131. 240. Two functions 'l/J and if; such that their difference is Lebesgue integrable

and yet

S ~f {(x, y) : if;(x) '5, y '5, 'l/J(x), x E [0, I]}

is not Lebesgue measurable. Exercise 2.2.1.16. 132.

241. A nonmeasurable continuous image of a null set (second example). Example 2.2.2.1. 132.

242. Any two Cantor-like sets are homeomorphic. Remark 2.2.2.1. 133.

Guide xxv

243. A nonmeasurable composition of a measurable function and a contin­uous strictly monotone function.

Exercise 2.2.2.1. 133. 244. The composition of a function of bounded variation and a measurable

function is measurable. Exercise 2.2.2.2. 133.

245. Egoroff's theorem. THEOREM 2.2.2.1. 133.

246. Counterexamples to weakened versions of Egoroff's theorem. Examples 2.2.2.2. 133, 2.2.2.3. 134.

247. Relations among modes of convergence. Exercises 2.2.2.3. 135, 2.2.2.4. 135, 2.2.2.5. 135, 2.2.2.6. 135,

2.2.2.7. 136, 2.2.2.8. 136, Example 2.2.2.4. 136. 248. A counterexample to a weakened version of the Radon-Nikodym theo­

rem. Exercise 2.2.2.9. 137.

249. The image measure catastrophe. Examples 2.2.2.5. 137, 2.2.2.6. 138.

250. A bounded semicontinuous function that is not equal almost every­where to any Riemann integrable function.

Exercise 2.2.2.10. 138, Note 2.2.2.1. 139. 251. A Riemann integrable function f and a continuous function g such

that fog is not equal almost everywhere to any Riemann integrable function.

Exercise 2.2.2.11. 139. 252. A continuous function of a Riemann integrable resp. Lebesgue mea­

surable function is Riemann integrable resp. Lebesgue measurable. Exercise 2.2.2.12. 139.

253. A differentiable function with a derivative that is not equal almost everywhere to any Riemann integrable function.

Example 2.2.2.7. 139. 254. A function that is not Lebesgue integrable and has a finite improper

Riemann integral. Exercise 2.2.2.13. 140.

255. If Rn l 00 there is in L1 (R., R.) a sequence {fn}nEN of nonnegative functions converging uniformly and monotonely to zero and such that for n in N,

Exercise 2.2.2.14. 140. 256. Fubini's and Tonelli's theorems.

pages 140-141.

xxvi Guide

257. Counterexamples to weakened versions of Fubini's and Tonelli's theo­rems.

Examples 2.2.2.8. 141. 258. A measurable function for which the graph has infinite measure.

Exercise 2.2.2.15. 142.

259. In ]RIR2 a function that is not Lebesgue integrable and for which both iterated integrals exist and are equal.

Example 2.2.2.9. 142.

260. In ]RIR2 a function that is not Riemann integrable and for which both iterated integrals exist and are equal.

Remark 2.2.2.2. 142. 261. Criteria for Lebesgue measurability of a function.

Exercise 2.2.2.16. 143. 262. Inadequacy of weakened criteria for measurability.

Exercise 2.2.2.17. 143. 263. A group invariant measure.

Example 2.2.3.1. 144. 264. The group 80(3) is not abelian.

Example 2.2.3.2. 145. 265. The Banach-Tarski paradox.

pages 144-156. 266. The number five in the Robinson version of the Banach-Tarski paradox

is best possible. THEOREM 2.2.3.4. 155, Exercise 2.2.3.8. 155.

Topological Vector Spaces

267. In an infinite-dimensional Banach space no Hamel basis is a (Schauder) basis.

Exercise 2.3.1.1. 156. 268. The Davie-Enflo example.

pages 157-8. 269. The trigonometric functions do not constitute a (Schauder) basis for

C (1l', C). Note 2.3.1.1. 158.

270. A nonretrobasis. Example 2.3.1.1. 159.

271. In l2 a basis that is not unconditional. Example 2.3.1.2. 160.

272. For a measure situation (X, S, JL) and an infinite orthonormal system {<Pn }nEN in eX, where limn -+oo <Pn (x) exists it is zero a.e.

THEOREM 2.3.1.1. 160.

Guide xxvii

273. If -00 < a < b < 00, {<Pn}nEN is an infinite orthonormal system in L2 ([a, b], JR), and sUPnEN l<pn(a)1 < 00 then limsuPnEN var(<pn) = 00.

COROLLARY 2.3.1.1. 161. 274. Phenomena related to THEOREM 2.3.1.1 and COROLLARY 2.3.1.1.

Exercise 2.3.1.2. 161, Example 2.3.1.3. 162.

275. A maximal biorthogonal set {xn' X~}nEN such that {Xn}nEN is not a basis.

Example 2.3.1.4. 162. 276. Banach spaces that are not the duals of Banach spaces.

Example 2.3.2.1. 163, Exercises 2.3.2.1. 163, 2.3.2.2. 163, Remark 2.3.2.1. 163, Example 2.3.2.2. 163.

277. In LP (JR, JR) an equivalence class containing no continuous function. Exercise 2.3.2.3. 163.

278. A separable Banach space for which the dual space is not separable. Example 2.3.2.3. 164.

279. A nonreflexive Banach space that is isometrically isomorphic to its second dual.

Example 2.3.2.4. 164. 280. In Cp (JR, JR) a dense set of infinitely differentiable functions.

Example 2.3.3.1. 165. 281. In Cp (JR, JR) a dense set of nowhere differentiable functions

Example 2.3.3.2. 165. 282. In C (T, JR) the set of nowhere differentiable functions is dense and of

the second category; its complement is dense and of the first category. THEOREM 2.3.3.1. 166, Exercise 2.3.3.1. 166.

283. In a normed infinite-dimensional vector space B there are arbitrarily large numbers of pairwise disjoint, dense, and convex subsets the union of which is B and for which B is their common boundary.

THEOREM 2.3.3.2. 167 through Exercise 2.3.3.5. 168. 284. Separability is a QL property.

Exercise 2.3.3.6. 168. 285. Noninclusions among the LP spaces.

Example 2.3.4.1. 170. 286. A linear function space that is neither an algebra nor a lattice.

Exercise 2.3.4.1. 170. 287. A linear function space that is an algebra and not a lattice.

Exercise 2.3.4.2. 170. 288. A linear function space that is a lattice and not an algebra.

Exercise 2.3.4.3. 170. 289. The set of functions for which the squares are Riemann integrable is

not a linear function space. Exercise 2.3.4.4. 170.

290. The set of functions for which the squares are Lebesgue integrable is not a linear function space.

Exercise 2.3.4.5. 170.

xxviii Guide

291. The set of semicontinuous functions is not a linear function space. Example 2.3.4.2. 171.

292. The set of periodic functions is not a linear function space. Exercise 2.3.4.6. 171.

293. A linear function space with two different norms such that the unit ball for one norm is a subset of the unit ball for the other and the difference set is norm dense in the larger ball.

Example 2.3.4.3. 171.

Topological Algebras

294. The algebra C~oo) (JR, q can be a topological algebra but cannot be a Banach algebra.

Example 2.4.1.1. 172, Note 2.4.1.1. 174, Exercise 2.4.1.1. 174.

295. Semisimplicity is a Q L property. Example 2.4.2.1. 174.

296. Semisimplicity is not a homomorphism invariant. Example 2.4.2.2. 175, Note 2.4.2.1. 175.

297. A radical algebra. Example 2.4.2.3. 175.

Differential Equations

298. Wronski's criterion for linear independence. THEOREM 2.5.1.1. 177.

299. A counterexample to a weakened version of Wronski's criterion. Exercise 2.5.1.1. 177.

300. An existence/uniqueness theorems for differential equations. THEOREM 2.5.2.1. 178.

301. A differential equation with two different solutions passing through a point.

Exercise 2.5.2.1. 178. 302. Rubel's example of superbifurcation.

Example 2,5.2.1. 178. 303. Lewy's example of a partial differential equation lacking even a distri­

bution solution. Example 2.5.2.2. 179.

304. A counterexample to a weakened version of the Cauchy-Kowalewski theorem.

Example 2.5.2.3. 180, Note 2.5.2.1. 180.

Guide xxix

Complex Variable Theory

305. Morera's theorem. THEOREM 2.6.1.1. 180.

306. A counterexample to a weakened version of Morera's theorem. Exercise 2.6.1.1. 180.

307. A power series for which the boundary of the circle of convergence is a natural boundary for the associated function.

Exercise 2.6.2.1. 181.

308. For a given closed subset F of T R ~f {z : z E c, Izl = R} a function holomorphic in D(O, R)O and for which the set SR(f) of singularities on TR is F.

Example 2.6.2.1. 181. 309. A function f a) holomorphic in D(O, 1)°, b) having T as its natural

boundary, and c) represented by a power series converging uniformly in D(O, 1).

Example 2.6.2.1. 182. 310. Functions a) holomorphic in D(O, 1)0, b) having T as natural boundary,

c) represented by power series converging uniformly in D(O, 1), and d) such that their values on T are infinitely differentiable functions of the angular parameter () used to describe T.

Examples 2.6.2.2. 182, 2.6.2.3. 182, Exercise 2.6.2.2. 182. 311. A region n that is not simply connected and in which a nonconstant

holomorphic function has a holomorphic square root. Example 2.6.3.1. 182.

312. A counterexample to the WeierstraB approximation theorem for C­valued functions.

Example 2.6.4.1. 183. 313. Every function in H (D(O, 1)°) n C (D(O, 1), C) is the limit of a uni­

formly convergent sequence of polynomials. Exercise 2.6.4.1. 183.

314. A counterexample to a weakened version of Rouche's theorem. Example 2.6.5.1. 184, Remark 2.6.5.1. 184.

315. De Brange's resolution of the Bieberbach-Robertson-Milin conjectures. pages 184-5.

316. A counterexample to a weakened version of the Bieberbach conjecture. Example 2.6.6.1. 185.

xxx Guide

The Euclidean Plane

317. Counterexamples for the parallel axiom. Examples 3.1.1.1. 187, 3.1.1.2. 187.

318. Desargue's theorem. THEOREM 3.1.1.1. 187.

319. Moulton's plane. Example 3.1.1.3. 188.

320. Nonintersecting connected sets that "cross." Example 3.1.2.1. 190.

321. A simple arc-image is nowhere dense in the plane. Exercise 3.1.2.1. 191.

322. A connected but not locally connected set. Example 3.1.2.2. 191.

323. Rectifiable and nonrectifiable simple arcs. Exercise 3.1.2.2. 191.

324. A nowhere differentiable simple arc. Example 3.1.2.3. 192.

325. An arc-image that fills a square. Example 3.1.2.4. 192.

326. An arc-image containing no rectifiable arc-image. Exercise 3.1.2.3. 193.

327. A function f for which the graph is dense in lR? Example 3.1.2.5. 193, Exercise 3.1.2.4. 193.

328. A connected set that becomes totally disconnected upon the removal of one of its points.

Example 3.1.2.6. 193.

329. For n in N, in R.2 n pairwise disjoint regions 'R1o "" 'Rn having a compact set F as their common boundary.

pages 195-198. 330. Aspects of the four color problem.

Note 3.1.2.1. 198. 331. Non-Jordan regions in lR?

Example 3.1.2.7. 198. 332. A non-Jordan region that is not the interior of its closure.

Example 3.1.2.8. 198.

Guide xxxi

Topological spaces

333. A sequence {Fn}nEN of bounded closed sets for which the intersection is empty.

Exercise 3.2.1.1. 198. 334. A nonconvergent Cauchy sequence.

Exercise 3.2.1.2. 199. 335. Cauchy completeness is not a topological invariant.

Note 3.2.1.2. 199. 336. In a complete metric space, a decreasing sequence of closed balls for

which the intersection is empty. Exercise 3.2.1.3. 199.

337. In a metric space an open ball that is not dense in the concentric closed ball of the same radius.

Exercise 3.2.1.4. 200. 338. In a metric space two closed balls such that the ball with the larger

radius is a proper subset of the ball with the smaller radius. Exercise 3.2.1.5. 200.

339. Topological spaces in which no point is a closed set and in which every net converges to every point.

Example 3.2.2.1. 200. 340. A topological space containing a countable dense set and a subset in

which there is no countable dense set. Exercise 3.2.2.1. 200.

341. A topological space containing a countable dense set and an uncount­able subset with an inherited discrete topology.

Exercise 3.2.2.2. 201. 342. Nonseparable spaces containing countable dense s).lbsets.

Exercises 3.2.2.2. 201, 3.2.2.3. 201. 343. The failure of the set of convergent sequences to define a topology.

Exercise 3.2.2.4. 201. 344. In topological vector spaces the distinctions among standard topolo­

gies. Exercise 3.2.2.5. 202.

345. The equivalence of weak sequential convergence and norm-convergence in ll.

Exercise 3.2.2.6. 202. 346. The "moving hump."

Remark 3.2.2.1. 202. 347. A sequence having a limit point to which no subsequence converges.

Exercise 3.2.2.7. 202. 348. Properties of the unit ball in the dual of a Banach space.

Remark 3.2.2.2. 203. 349. A continuous map that is neither open nor closed.

Exercise 3.2.2.8. 203.

xxxii Guide

350. A map that is open and closed and not continuous. Exercise 3.2.2.9. 203.

351. A closed map that is neither continuous nor open. Exercise 3.2.2.9. 203.

352. A map that is continuous and open but not closed. Exercise 3.2.2.10. 203.

353. An open map that is neither continuous nor closed. Exercise 3.2.2.11. 203.

354. A map that is continuous and closed but not open. Exercise 3.2.2.12. 203.

355. Two nonhomeomorphic spaces each of which is the continuous bijective image of the other.

Example 3.2.2.2. 204. 356. Wild spheres in ]R3.

Figures 3.2.2.2. 206, 3.2.2.3. 207. 357. Antoine's necklace.

Figure 3.2.2.4. 207.

Exotica in Differential Topology

358. Homeomorphic nondiffeomorphic spheres. Example 3.3.1. 208.

359. There are uncountably many nondiffeomorphic differential geometric structures for ]R4.

page 208. 360. The resolution of the Poincare conjecture in ]Rn, n i 3.

pages 208-9.

Independence in Probability

361. For independent random variables the integral of the product is the product of the integrals.

Exercise 4.1.1. 211. 362. A probability situation where there are only trivial instances of inde­

pendence. Exercise 4.1.2. 211.

363. Pairwise independence does not imply independence. Example 4.1.1. 212.

364. Compositions of Borel measurable functions and independent random variables.

Exercise 4.1.3. 212.

Guide xxxiii

365. Random variables independent of no nontrivial random variables. Example 4.1.2. 212, Note 4.1.1. 213.

366. The metric density theorem. page 213.

367. Independent random variables cannot span a Hilbert space of dimen­sion less than three.

THEOREM 4.1.1. 214. 368. In THEOREM 4.1.1 three is best possible.

Remark 4.1.1. 215. 369. The Rademacher functions constitute a maximal set of independent

random variables. Exercise 4.1.4. 215.

370. A general construction of a maximal family of independent random variables.

Example 4.1.3. 215.

Stochastic Processes

371. If f and 9 are independent and if f ±g are independent then f, g, f ±g are all normally distributed.

LEMMAS 4.2.1. 218, 4.2.2. 218. 372. The nonexistence of a Gauf3ian measure on Hilbert space.

LEMMA 4.2.3. 220. 373. The nonexistence of a nontrivial translation-invariant or unitarily in­

variant measure on Hilbert space. Example 4.2.1. 220.

Transition matrices

374. For a transition matrix P a criterion for the existence of limn --+ oo pn. THEOREM 4.3.1. 222.

375. The set 'P of n x n transition matrices as a set in the nonnegative orthant jR(n2 ,+).

Exercise 4.3.1. 222. 376. The set 'P 00 of n x n transition mattices P such that limn --+oo pn exists

is a null set (An2_n) and 'P\ 'Poo is a dense open subset of 'P. Exercise 4.3.2. 222.

xxxiv Guide

Logic

377. Godel's completeness theorem. page 225.

378. Godel's count ability theorem. page 225.

379. The Lowenheim-Skolem theorem. page 226.

380. Godel's incompleteness (undecidability) theorem. page 226.

381. Computability and the halting problem. pages 226-8.

382. Hilbert's tenth problem. page 228.

383. The Boolos-Vesley discussion of Godel's incompleteness theorem. Note 5.1.4. 229.

Set Theory

384. The consistency of the Continuum Hypothesis. page 230.

385. The independence of the Axiom of Choice and the Generalized Con­tinuum Hypothesis.

page 230. 386. Solovay's axiom and functional analysis.

pages 230-1.