# Real Mathematical Analysis

Springer Science & Business Media, Nov 14, 2003 - Mathematics - 440 pages
Was plane geometry your favorite math course in high school? Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is pure mathematics, and I hope it appeals to you, the budding pure mathematician. Berkeley, California, USA CHARLES CHAPMAN PUGH Contents 1 Real Numbers 1 1 Preliminaries 1 2 Cuts . . . . . 10 3 Euclidean Space . 21 4 Cardinality . . . 28 5* Comparing Cardinalities 34 6* The Skeleton of Calculus 36 Exercises . . . . . . . . 40 2 A Taste of Topology 51 1 Metric Space Concepts 51 2 Compactness 76 3 Connectedness 82 4 Coverings . . . 88 5 Cantor Sets . . 95 6* Cantor Set Lore 99 7* Completion 108 Exercises . . . 115 x Contents 3 Functions of a Real Variable 139 1 Differentiation. . . . 139 2 Riemann Integration 154 Series . . 179 3 Exercises 186 4 Function Spaces 201 1 Uniform Convergence and CO[a, b] 201 2 Power Series . . . . . . . . . . . . 211 3 Compactness and Equicontinuity in CO . 213 4 Uniform Approximation in CO 217 Contractions and ODE's . . . . . . . . 228 5 6* Analytic Functions . . . . . . . . . . . 235 7* Nowhere Differentiable Continuous Functions . 240 8* Spaces of Unbounded Functions 248 Exercises . . . . . 251 267 5 Multivariable Calculus 1 Linear Algebra . . 267 2 Derivatives. . . . 271 3 Higher derivatives . 279 4 Smoothness Classes . 284 5 Implicit and Inverse Functions 286 290 6* The Rank Theorem 296 7* Lagrange Multipliers 8 Multiple Integrals . .

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### Contents

 I 1 II 10 III 21 IV 28 V 34 VI 36 VII 40 VIII 51
 XXXII 290 XXXIII 296 XXXIV 300 XXXV 313 XXXVI 325 XXXVII 334 XXXVIII 337 XXXIX 338

 IX 76 X 82 XI 88 XII 95 XIII 99 XIV 108 XV 139 XVI 154 XVII 179 XVIII 186 XIX 201 XX 211 XXI 213 XXII 217 XXIII 228 XXIV 235 XXV 240 XXVI 248 XXVII 267 XXVIII 271 XXIX 279 XXX 284 XXXI 286
 XL 339 XLI 340 XLII 342 XLIII 345 XLIV 363 XLV 367 XLVI 371 XLVII 376 XLVIII 383 XLIX 387 L 391 LI 396 LII 401 LIII 407 LIV 409 LVI 411 LVII 412 LVIII 413 LIX 414 LX 415 LXI 417 LXII 431 Copyright