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Suggested Reading 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 anal- ysis 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. © Springer International Publishing Switzerland 2015 467 C.C. Pugh, Real Mathematical Analysis, Undergraduate Texts in Mathematics, DOI 10.1007/978-3-319-17771-7 Bibliography 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 Dieudonn´e, 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 Gouvˆea, 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’int´egration 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. © Springer International Publishing Switzerland 2015 469 C.C. Pugh, Real Mathematical Analysis, Undergraduate Texts in Mathematics, DOI 10.1007/978-3-319-17771-7 470 Bibliography 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 -H¨older, 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 Arzel`a-Ascoli Propagation Theorem, 227 Arzel`a-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 © Springer International Publishing Switzerland 2015 471 C.C. Pugh, Real Mathematical Analysis, Undergraduate Texts in Mathematics, DOI 10.1007/978-3-319-17771-7 472 Index 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 additivitadditivity,y, 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 doppelg¨anger, 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 Fr´echet derivative, 284 identity map, 31 front face,face, 343343 Identity Theorem for analytic functions, Fubini’s Theorem, 316 268
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