Suggested Reading

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Suggested Reading

There are many books on more advanced analysis and topology. Among my favorites in the “not too advanced” category are these. 1. Kenneth Falconer, The Geometry of Fractal Sets. Here you should read about the Kakeya problem:Howmuch area is needed to reverse the position of a unit needle in the plane by a continuous motion? Falconer also has a couple of later books on fractals that are good. 2. Thomas Hawkins, Lebesgue’s Theory of Integration. You will learn a great deal about the history of Lebesgue integration and analysis around the turn of the last century from this book, including the fact that many standard attributions are incorrect. For instance, the Cantor set should be called the Smith set; Vitali had many of the ideas credited solely to Lebesgue, etc. Hawkins’ book is a real gem. 3. John Milnor, Topology from the Differentiable Viewpoint. Milnor is one of the clearest mathematics writers and thinkers of the twentieth century. This is his most elementary book, and it is only seventy-six pages long. 4. James Munkres, Topology, a First Course. This is a first-year graduate text that deals with some of the same material you have been studying. 5. Robert Devaney, An Introduction to Chaotic Dynamical Systems. This is the book you should read to begin studying mathematical dynamics. It is first rate. One thing you will observe about all these books – they use pictures to convey the mathematical ideas. Beware of books that don’t.

1. Ralph Boas, A Primer of Real Functions, The Mathematical Association of America, Washington DC, 1981. 2. Andrew Bruckner, Differentiation of Real Functions, Lecture Notes in Mathe- matics, Springer-Verlag, New York, 1978. 3. John Burkill, The Lebesgue Integral,Cambridge University Press, London, 1958. 4. Paul Cohen, Set Theory and the Continuum Hypothesis, Benjamin, New York, 1966. 5. Robert Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin Cummings, Menlo Park, CA, 1986. 6. Jean Dieudonné, Foundations of Analysis, Academic Press, New York, 1960. 7. Kenneth Falconer, The Geometry of Fractal Sets,Cambridge University Press, London, 1985. 8. Russell Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock,The American Mathematical Society, Providence, RI, 1994. 9. Fernando Gouvêa, p-adic Numbers, Springer-Verlag, Berlin, 1997. 10. Thomas Hawkins, Lebesgue’s Theory of Integration, Chelsea, New York, 1975. 11. George Lakoff, Where Mathematics Comes From, Basic Books, New York, 2000. 12. Edmund Landau, Foundations of Analysis, Chelsea, New York, 1951. 13. Henri Lebesgue, Le¸cons sur l’intégration et la recherche des fonctions primi- tives, Gauthiers-Villars, Paris, 1904. 14. John Littlewood, Lectures on the Theory of Functions, Oxford University Press, Oxford, 1944. 15. Ib Madsen and Jørgen Tornehave, From Calculus to Cohomology,Cambridge University Press, Cambridge, 1997.

16. Jerrold Marsden and Alan Weinstein, Calculus III, Springer-Verlag, New York, 1998. 17. Robert McLeod, The Generalized Riemann Integral, The Mathematical Asso- ciation of America, Washington DC, 1980. 18. John Milnor, Topology from the Differentiable Viewpoint, Princeton University Press, Princeton, 1997. 19. Edwin Moise, Geometric Topology in Dimensions 2 and 3, Springer-Verlag, New York, 1977. 20. James Munkres, Topology, a First Course, Prentice Hall, Englewood Cliffs, NJ, 1975. 21. Murray Protter and Charles Morrey, A First Course in Real Analysis, Springer- Verlag, New York, 1991. 22. Dale Rolfsen, Knots and Links, Publish or Perish, Berkeley, 1976. 23. Halsey Royden, Real Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1988. 24. Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, 1976. 25. James Stewart, Calculus with Early Transcendentals, Brooks Cole, New York, 1999. 26. Arnoud van Rooij and Wilhemus Schikhof, A Second Course on Real Functions, Cambridge University Press, London, 1982. Index

C1 Mean Value Theorem, 289 abuse of notation, 7 Cr M-test, 297 accumulation point, 92 r C equivalence, 302 address string, 107 r C norm, 296 adheres, 65 Fσ-set, 201, 395 aleph null, 31 Gδ-set, 201, 395 algebraic number, 51 1 L -convergence, 464 almost every, 407 L1 -norm, 464 almost everywhere, 175 α -Hölder, 265 alternating harmonic series, 196 δ -dense, 265 alternating multilinear functional, 352 -chain, 131 alternating series, 195 -principle, 21 ambiently diffeomorphic, 378 σ-algebra, 389 ambiently homeomorphic, 115 σ-compact, 262 ∗ analytic, 158, 248 σ -compact, 268 Analyticity Theorem, 250 f-translation, 411 Antiderivative Theorem, 185, 431 k-chain, 342 p-adic metric, 136 Antoine’s Necklace, 117 p-series, 194 arc, 131 r-neighborhood, 68 area of a rectangle, 384 t-advance map, 246 argument by contradiction, 8 (, δ)-condition, 65 Arzelà-Ascoli Propagation Theorem, 227 Arzelà-Ascoli Theorem, 224 absolute continuity, 429 ascending k-tuple, 333 absolute convergence, 192, 217 associativity, 14, 335 absolute property, 85 average derivative, 289 abstract outer measure, 389 Average Integral Theorem, 426

Baire class 1, 201 cell, 328 Baire’s Theorem, 256 center of a starlike set, 130 balanced density, 422, 458 chain connected, 131 Banach Contraction Principle, 240 Chain Rule, 150, 285 Banach indicatrix, 460 Change of Variables Formula, 319 Banach space, 296 characteristic function, 171 basic form, 331 Chebyshev Lemma, 434 Bernstein polynomial, 229 class Cr, 158, 295 bijection, 31 class C∞, 295 bilinear, 287 clopen, 67 block test, 208 closed form, 347 Bolzano-Weierstrass Theorem,80 closed neighborhood, 94 Borel measurability, 443 closed set, 66 Borel’s Lemma, 267 closed set condition, 72 boundary, 92, 141 closure, 70, 92 k boundary of a -cell, 343 cluster point, 92, 140 bounded above, 13 co-Cauchy, 119 bounded function, 98, 261 codomain, 30 bounded linear transformation, 279 coherent labeling, 110 bounded metric, 138 common refinement, 168 bounded set, 97 commutative diagram, 302 bounded variation, 438 compact, 79 box, 26 comparable norms, 366 Brouwer Fixed-Point Theorem, 240, 353 Comparison Test, 192 bump function, 200 complement, 45 Cantor function, 186 complete, 14, 78 Cantor Partition Lemma, 113 completed undergraph, 407 Cantor piece, 112 Completion Theorem, 119 Cantor set, 105 complex analytic, 251 Cantor space, 112 complex derivative, 360 Cantor Surjection Theorem, 108 composite, 31 cardinality, 31 concentration, 422 Cauchy completion, 122 condensation point, 92, 140 Cauchy condition, 18, 77 condition number, 361 Cauchy Convergence Criterion, 19, 191 conditional convergence, 192, 464 Cauchy product, 210 cone map, 349 Cauchy sequence, 77 cone on a metric space, 139 Cauchy-Binet Formula, 339, 363 connected, 86 Cauchy-Riemann Equations, 360 connected component, 147 Cauchy-Schwarz Inequality, 23 conorm, 281, 366 Cavalieri’s Principle, 318, 414 continuity in a metric space, 61 Index 473 continuously differentiable, 157 differential 1-form, 327 Continuum Hypothesis, 31, 137, 145 differential quotient, 149 contraction, weak contraction, 240, 266 differentiation past the integral, 290 convergence, 18, 60, 191 dipole, 343 convex, 26 directional derivative, 369 convex combinations, 27, 49 disconnected, 86 convex function, 49 discontinuity of the first, second kind, 204 convex hull, 115 discrete metric, 58 countable, 31 disjoint, 2 ccountableountable additivity,additivity, countable subaddi- distance from a point to a set, 130 tivity, 384,384,38 3899 distance function, 58 countable base, 141 divergence of a series, 191 counting measure, 450 divergence of a vector field, 346 covering, 98 division of a metric space, 109 covering compact, 98 domain, 29 critical point, critical value, 204, 459 Dominated Convergence Theorem, 409 cube, 26 domination of one series by another, 192 Cupcake Theorem, 145 doppelgänger, 442 curl, 347 dot product, 22 double density point, 465 Darboux continuous, 154 duality equation, 338 Darboux integrable, Darboux integral, 167 dyadic, 47 de Rham cohomology, 352 dyadic ruler function, 204 De Morgan’s Law, 45 Dedekind cut, 12 Egoroff’s Theorem, 448 Denjoy integral, 464 embedding, 85 dense, 107 empty set, 2 density point, 422 envelope sequences, 408 denumerable, 31 equicontinuity, 224 derivate, 434 equivalence relation, equivalence class, 3 derivative, 149 Euler characteristic, 50 derivative (multivariable), 282 Euler’s Product Formula, 210 derivative growth rate, 248 exact form, 347 determinant, 363 exponential growth rate, 194 Devil’s ski slope, 188, 456 extension of a function, 129 Devil’s staircase function, 186 exterior derivative, 337 diagonalizable matrix, 368 diameter in a metric space, 82 fat Cantor set, 108, 203 diffeomorphism, 163, 300 Fatou’s Lemma, 410 differentiability of order r, 157 field, 16 differentiable (multivariable), 282 finite, 31 differentiable function, 149, 151 finite additivity, 390 474 Index

finite intersection property, 134 hull, 400 fixed-point, 47, 240 hyperspace, 144 flow, 246 flux, 346 idempotent, 70 Fréchet derivative, 284 identity map, 31 front face,face, 343343 Identity Theorem for analytic functions, Fubini’s Theorem, 316 268 Fubini-Tonelli Theorem, 416 image, 30 function, 29 implicit function, 297 function algebra, 234 Implicit Function Theorem, 298 functional, 328 improper Riemann integral, 191 Fundamental Theorem of Calculus, 183, inclusion cell, 334 426 indicator function, 171 Fundamental Theorem of Continuous Func- infimum, 17 tions, 41 infinite, 31 infinite address string, 107 gap interval, 108, 112 infinite product, 209 Gauss Divergence Theorem, 346 infinitely differentiable, 157 generalgeneral k-f-form,orm, 331331 Inheritance Principle, 73, 74 generalgenerall linearinear group,group, 372372 inherited metric, 58 Generalized Heine-Borel Theorem, 103 inherited topology, 74 generic, 256 initial condition for an ODE, 242 geometric series, 191 injection, 30 gradient, 311 inner measure, 384 grand intersection, 134 inner product, inner product space, 28 greatest lower bound, 47 integer lattice, 24 Green’s Formula, 346 Integral Test, 193 growing steeple, 214 integrally equivalent, 205 integration by parts, 189 Hölder condition, 198 integration by substitution, 189 Hahn-Mazurkiewicz Theorem, 143 interior, 92, 140 Hairy Ball Theorem, 381 Intermediate Value Theorem, 40 harmonic series, 192 Intermediate Value Theorem for f , 154 Hausdorff metric, 144 intrinsic property, 85 Hawaiian earring, 132 Inverse Function Theorem, 162, 301 Heine-Borel Theorem, 80, 81 inverse image, 71 Heine-Borel Theorem in a Function Space, isometry, isometric, 126 228 iterate, 138 Higher Order Chain Rule, 374 Higher Order Leibniz Rule, 199 Jacobian, 319, 329 Hilbert cube, 143 Jordan content, 319, 450, 451 homeomorphism, 62 Jordan Curve Theorem, 144 Index 475

Jordan measurable, 451 magnitude of a number, of a vector, 16, jump, jump discontinuity, 49, 204 23 Manhattan metric, 76 kernel, 401 manimanifold,fold 345345 map, mapping, 29 L’Hôpital’s Rule, 153 maximum stretch, 279 Lagrange form of the Taylor remainder, meager subset, 256 160 mean value property, 151 Lagrange multiplier, 310 Mean Value Theorem, 151, 288 least upper bound, 13 measurability, measure, 389 Least Upper Bound Property, 14 measurable function, 406 Lebesgue Density Theorem, 422 Measurable Product Theorem, 402 Lebesgue Dominated Convergence Theo- measurable with respect to an outer mea- rem, 409 sure, 389 Lebesgue integrability, Lebesgue integral, Measure Continuity Theorem, 392 406 measure continuous, 429429 Lebesgue measurability, Lebesgue measure, measure space, 393 389 measure theoretictheoretic boundary,boundary,40 4011 Lebesgue Monotone Convergence Theo- measure-theoretic connectedness, 465 rem, 407 Mertens’ Theorem, 210 Lebesgue number, 100 mesemorpmesemorphism,hism, 393 meseomorpmeseomorphism,hism, 393393 Lebesgue outer measure, 383 mesh of a partition, 164 Lebesgue’s Antiderivative Theorem, 431 mesmesisometry,isometry, 393393 Lebesgue’s Fundamental Theorem of Cal- metric space, metric subspace, 57, 58 culus, 426 middle-quarters Cantor set, 203 Lebesgue’s Main Theorem, 430, 439 middle-thirds Cantor set, 105 Leibniz Rule, 149, 285 minimum stretch, 366 length of a vector, 23 modulus of continuity, 264 length of an interval, 383 Monotone Convergence Theorem, 407 limit, 65 monotonicity, 125 limit point, 65 Moore-Kline Theorem, 112 limit set, 68 Morse-Sard Theorem, 204 linear transformation, 277 multilinear functional, 352 Lipeomorphism, 452 Lipschitz condition, 244 name of a form, 327 locally path connected, 143 natural numbers, 1 locally path-connected, 132 nearly continuous, 447 logarithm function, 186 nearly uniform convergence, 448 lower Lebesgue sum, 440 neighborhood, 70 lower sum, lower integral, 166 nested sequence, 81 Lusin’s Theorem, 447 norm, normed space, 28, 279 476 Index nowhere dense, 107 Rademacher’s Theorem, 206, 438 Radius of Convergence Theorem, 197 ODE, 242 range, 30 one-to-one, 30 rank, Rank Theorem, 301, 303 onto, 30 Ratio Mean Value Theorem, 152 open covering, 98 Ratio Test, 195 open mapping, 127 rational cut, 13 open set, 66 rational numbers, 2 open set condition, 72 rational ruler function, 173 operator norm, 279 real number, 12 orbit, 138, 441 rearrangement of a sequence, 126 ordered field, 16 rearrangement of a series, 209 orthant, 24 reduction of a covering, 98 oscillating discontinuity, 205 Refinement Principle, 168 oscillation, 177 regularity hierarchy, 158 outer measure, 383 regularity of Lebesgue measure, 395 regularity sandwich, 396 parallelogram law, 53 retraction, 353 partial derivative, 284 Riemann ζ-function, 210 partial product, 209 Riemann integrability, Riemann integral, partial sum, 191 164 partition, 113 Riemann measurable, 319 partition pair, 164 Riemann sum, 164 patches, 99 Riemann’s Integrability Criterion, 171 path, path-connected, 90 Riemann-Lebesgue Theorem, 175 Peano curve, 112 Root Test, 194 Peano space, 143 perfect, 94 sample points, 164 Picard’s Theorem, 244 Sandwich Principle, 173 piece of a compact metric space, 109 Sard’s Theorem, 204 piecewise continuous function, 172 satellite, 458 Poincaré Lemma, 348 sawtooth function, 254 pointwise convergence, pointwise limit, 211 Schroeder-Bernstein Theorem, 36 pointwise equicontinuity, 224, 261 scraps, 99 Polar Form Theorem, 362 second derivative, 291 positive definiteness, 58 separable metric space, 141 preimage, 71 separates points (function algebra), 234 preimage measurability, 416 separation, 86 proper subset, 86 sshadowhadow area, 329 pullback, pushforward, 338 shear matrix, 320, 368 sign of a permutation, 363 quasi-round, 449 signed area, 330 Index 477 signed commutativity, 331 Term by Term Integration Theorem, 219 simple closed curve, 144 thick and thin subsets, 256 simple form, 331 topological equivalence, 73 simple function, 456 topological property, 71 simple region, 377 topological space, 67 simply connected, 347 topologist’s sine circle, 132 singleton set, 2 topologist’s sine curve, 91 slice, 316, 403, 414 total derivative, 284 sslicelice integral, 313166 total length of a covering, 108, 175, 384 sliding secant method, 155 total undergraph, 453 slope over an interval, 434 total variation of a function, 438 smooth, 157, 295 totally bounded, 103 solution of an ODE, 242 totally disconnected, 105 somewhere dense, 107 trajectory of a vector field, 243 space-filling, 112 transcendental number, 51 spherical shell, 379 transformation, 29 staircase curve, 376 Triangle Inequality, 16 starlike, 130, 351 Triangle Inequality for distance, 24 steeple functions, 214 Triangle Inequality for vectors, 24 Steinhaus’ Theorem, 441 trichotomy, 16 step function, 172 trigonometric polynomial, 238 Stokes’ Curl Theorem, 347 truncation of an address, 107 Stokes’ Formula for a Cube, 343 Stokes’ Formula for a general cell, 345 ultrametric, 136 Stone-Weierstrass Theorem, 234 unbounded set, 97 subcovering, 98 uncountable, 31 subfield, 16 undergraph, 164, 406 r sublinear, 282 uniform C convergence, 295 subsequence, 60 uniform continuity, 52, 85 sup norm, 214 uniform convergence, 211, 217 support of a function, 200 uniform equicontinuity, 261 supremum, 17 unit ball, sphere, 26 surjection, 30 unit cube, 26 universal compact metric space, 108 tail of a series, 192 upper semicontinuity, 147, 275, 454 tame, 116 upper sum, upper integral, 166 target, 30 utility problem, 144 taxicab metric, 76 Taylor Approximation Theorem, 160 vanishing at a point (function algebra), Taylor polynomial, 159 234 Taylor series, 161, 248 vector field, 243, 346 Taylor’s Theorem, 251 vector ODE, 242 478 Index

Vitali covering, 418 Vitali Covering Lemma, 418, 422 Volume MultiplierMultiplier Formula, 320320 wedge product, 334 Weierstrass Approximation Theorem, 228 Weierstrass M-test, 217 wild, 117

Zeno’s staircase function, 174 zero locus, 268, 461 zero set, 108, 175, 315, 386 Zero Slice Theorem, 403 zeroth derivative, 157