Vedanta and Advaita Saivagama of Kashmir: A Comparative Study

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. 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

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.

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. -- Publisher description Explains set theory, sequences, continuity, differentiation, integrals, and vector-space concepts