Real Mathematical Analysis

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Springer Science & Business Media, Mar 1, 2002 - Mathematics - 437 pages
3 Reviews
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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Great introductory book, especially for people who wish to self-study Real Analysis. Note that for any high school students who wish to self-study from this book should have a strong background in Mathematics. General understanding of the Propositional Calculus (a.k.a. Mathematical Logic) and Set Theory will help in the beginning, but once you get through the first chapter, the rest of the book slowly becomes accessible.  

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
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