Principles of Mathematical Analysis, Third Edition

✍ Scribed by Walter Rudin

• Publisher: McGraw-Hill Science/Engineering/Math
• Year: 1976
• Tongue: English
• Leaves: 351
• Edition: 3rd
• Category: Library

✦ Synopsis

The third edition of this well known text continues to provide a solid foundation in mathematical analysis for undergraduate and first-year graduate students. The text begins with a discussion of the real number system as a complete ordered field. (Dedekind's construction is now treated in an appendix to Chapter I.) The topological background needed for the development of convergence, continuity, differentiation and integration is provided in Chapter 2. There is a new section on the gamma function, and many new and interesting exercises are included. This text is part of the Walter Rudin Student Series in Advanced Mathematics.

✦ Table of Contents

CONTENTS......Page 4
PREFACE......Page 8
THE REAL AND THE COMPLEX NUMBER SYSTEMS
......Page 10
ORDERED SETS......Page 12
FIELDS......Page 14
THE REAL FIELD......Page 17
THE EXTENDED REAL NUMBER SYSTEM......Page 20
THE COMPLEX FIELD......Page 21
EUCLIDEAN SPACES......Page 25
APPENDIX......Page 26
EXERCISES......Page 30
FINITE, COUNTABLE, AND UNCOUNTABLE SETS......Page 33
COMPACT SETS......Page 45
METRIC SPACES......Page 39
PERFECT SETS......Page 50
CONNECTED SETS......Page 51
EXERCISES......Page 52
CONVERGENT SEQUENCES......Page 56
SUBSEQUENCES......Page 60
CAUCHY SEQUENCES......Page 61
UPPER AND LOWER LIMITS......Page 64
SOME SPECIAL SEQUENCES......Page 66
SERIES......Page 67
SERIES OF NONNEGATIVE TERMS......Page 70
THE NUMBER e......Page 73
THE ROOT AND RATIO TESTS......Page 74
POWER SERIES......Page 78
SUMMATION BY PARTS......Page 79
ABSOLUTE CONVERGENCE......Page 80
ADDITION AND MULTIPLICATION OF SERIES......Page 81
REARRANGEMENTS......Page 84
EXERCISES......Page 87
LIMITS OF FUNCTIONS......Page 92
CONTINUOUS FUNCTIONS......Page 94
CONTINUITY AND COMPACTNESS......Page 98
CONTINUITY AND CONNECTEDNESS......Page 102
DISCONTINUITIES......Page 103
MONOTONIC FUNCTIONS......Page 104
INFINITE LIMITS AND LIMITS AT INFINITY......Page 106
EXERCISES......Page 107
THE DERIVATIVE OF A REAL FUNCTION......Page 112
MEAN VALUE THEOREMS......Page 116
THE CONTINUITY OF DERIVATIVES
......Page 117
L'HOSPITAL'S RULE......Page 118
DERIVATIVES OF HIGHER ORDER......Page 119
DIFFERENTIATION OF VECTOR-VALUED FUNCTIONS......Page 120
EXERCISES......Page 123
DEFINITION AND EXISTENCE OF THE INTEGRAL......Page 129
PROPERTIES OF THE INTEGRAL
......Page 137
INTEGRATION AND DIFFERENTIATION......Page 142
INTEGRATION OF VECTOR-VALUED FUNCTIONS......Page 144
RECTIFIABLE CURVES......Page 145
EXERCISES......Page 147
DISCUSSION OF MAIN PROBLEM......Page 152
UNIFORM CONVERGENCE......Page 156
UNIFORM CONVERGENCE AND CONTINUITY......Page 158
UNIFORM CONVERGENCE AND INTEGRATION......Page 160
UNIFORM CONVERGENCE AND DIFFERENTIATION......Page 161
EQUICONTINUOUS FAMILIES OF FUNCTIONS......Page 163
THE STONE-WEIERSTRASS THEOREM......Page 168
EXERCISES......Page 174
POWER SERIES......Page 181
THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS......Page 187
THE TRIGONOMETRIC FUNCTIONS......Page 191
THE ALGEBRAIC COMPLETENESS OF THE COMPLEX FIELD......Page 193
THE GAMMA FUNCTION......Page 201
EXERCISES......Page 205
LINEAR TRANSFORMATIONS......Page 213
DIFFERENTIATION......Page 220
THE CONTRACTION PRINCIPLE......Page 229
THE INVERSE FUNCTION THEOREM......Page 230
THE IMPLICIT FUNCTION THEOREM......Page 232
THE RANK THEOREM......Page 237
DETERMINANTS......Page 240
DERIVATIVES OF HIGHER ORDER......Page 244
DIFFERENTIATION OF INTEGRALS......Page 245
EXERCISES......Page 248
INTEGRATION......Page 254
PRIMITIVE MAPPINGS......Page 257
PARTITIONS OF UNITY......Page 260
CHANGE OF VARIABLES......Page 261
DIFFERENTIAL FORMS......Page 262
SIMPLEXES AND CHAINS......Page 275
STOKES' THEOREM......Page 281
CLOSED FORMS AND EXACT FORMS......Page 284
VECTOR ANALYSIS......Page 290
EXERCISES......Page 297
SET FUNCTIONS......Page 309
CONSTRUCTION OF THE LEBESGUE MEASURE......Page 311
MEASURABLE FUNCTIONS......Page 319
SIMPLE FUNCTIONS......Page 322
INTEGRATION......Page 323
COMPARISON WITH THE RIEMANN INTEGRAL......Page 331
FUNCTIONS OF CLASS L^
2......Page 334
EXERCISES......Page 341
LIST OF SPECIAL SYMBOLS......Page 346
INDEX......Page 348
BIBLIOGRAPHY......Page 344