Real analysis

In the author's preface, he states that the prerequisites are "one semester of advanced calculus or real analysis at the undergraduate level". So, this book cannot be judged as an 'intro to real analysis'.I just want to comment on how I have experienced this book. Let me mention that I am using this for self-study after completing a course using Rudin's Principles of Mathematical Analysis (we covered every chapter except Ch. 10 on integration in R^n). I picked this up to review analysis with the goal of covering function spaces and measure theory with more emphasis that Rudin. This book does just that! But, I also wanted a book that stays in R for the Lebesgue measure. Having read the first 3 chapters of Folland, I didn't really think I 'understood' the material even though I could do the exercises (but not without a lot of sweat and coffee). (At one point I felt I became a function: [input] facts, assumptions then [output] proofs, ie hw exercises.) Folland does everything for abstract measures and treats the Lebesgue measure as a corollary. Having said that, this books hits the spot.A previous reviewer said this book was informal, unprofessional, and chatty. I do agree with him on that the book is very informal in the exposition and is chatty. I feel that this might be very distracting for those who do not wish to be specialists in analysis, or to those who are seeing analysis for the first time. However, for someone who has finished, say Baby Rudin, this book IS AMAZING. His chatty 'foreshadowing' is the best part, since by now you are trying to see the 'big picture'. In this respect, the chattiness tells of the shortcomings of the previous theory and points one to the right questions to ask.I think this book shines for the purpose of an intermediate course between Baby Rudin and graduate real analysis ala Folland. As such, the exercises are at the perfect level and include standard, important, and interesting results and extensions. I don't think this book is rigorous enough for a real course at the graduate level, however.A final note, the editorial (why?)'s placed throughout do get annoying but I feel they make sure you do not take results for granted, an all too common habit when reading advanced math. Aimed At Advanced Undergraduates And Beginning Graduate Students, Real Analysis Offers A Rigorous Yet Accessible Course In The Subject. Carothers, Presupposing Only A Modest Background In Real Analysis Or Advanced Calculus, Writes With An Informal Style And Incorporates Historical Commentary As Well As Notes And References. The Book Looks At Metric And Linear Spaces, Offering An Introduction To General Topology While Emphasizing Normed Linear Spaces. It Addresses Function Spaces And Provides Familiar Applications, Such As The Weierstrass And Stone-weierstrass Approximation Theorems, Functions Of Bounded Variation, Riemann-stieltjes Integration, And A Brief Introduction To Fourier Analysis. Finally, It Examines Lebesgue Measure And Integration On The Line. Illustrations And Abundant Exercises Round Out The Text. Real Analysis Will Appeal To Students In Pure And Applied Mathematics As Well As Researchers In Statistics, Education, Engineering, And Economics.--jacket. Part 1 Metric Spaces -- 1 Calculus Review 3 -- Real Numbers 3 -- Limits And Continuity 14 -- 2 Countable And Uncountable Sets 18 -- Equivalence And Cardinality 18 -- Cantor Set 25 -- Monotone Functions 31 -- 3 Metrics And Norms 36 -- Metric Spaces 37 -- Normed Vector Spaces 39 -- More Inequalities 43 -- Limits In Metric Spaces 45 -- 4 Open Sets And Closed Sets 51 -- Open Sets 51 -- Closed Sets 53 -- Relative Metric 60 -- 5 Continuity 63 -- Continuous Functions 63 -- Homeomorphisms 69 -- Space Of Continuous Functions 73 -- 6 Connectedness 78 -- Connected Sets 78 -- 7 Completeness 89 -- Totally Bounded Sets 89 -- Complete Metric Spaces 92 -- Fixed Points 97 -- Completions 102 -- 8 Compactness 108 -- Compact Metric Spaces 108 -- Uniform Continuity 114 -- Equivalent Metrics 120 -- 9 Category 128 -- Discontinuous Functions 128 -- Baire Category Theorem 131 -- Part 2 Function Spaces -- 10 Sequences Of Functions 139 -- Historical Background 139 -- Pointwise And Uniform Convergence 143 -- Interchanging Limits 150 -- Space Of Bounded Functions 153 -- 11 Space Of Continuous Functions 162 -- Weierstrass Theorem 162 -- Trigonometric Polynomials 170 -- Infinitely Differentiable Functions 176 -- Equicontinuity 178 -- Continuity And Category 183 -- 12 Stone-weierstrass Theorem 188 -- Algebras And Lattices 188 -- Stone-weierstrass Theorem 194 -- 13 Functions Of Bounded Variation 202 -- Functions Of Bounded Variation 202 -- Helly's First Theorem 210 -- 14 Riemann-stieltjes Integral 214 -- Weights And Measures 214 -- Riemann-stieltjes Integral 215 -- Space Of Integrable Functions 221 -- Integrators Of Bounded Variation 225 -- Riemann Integral 232 -- Riesz Representation Theorem 234 -- Other Definitions, Other Properties 239 -- 15 Fourier Series 244 -- Dirichlet's Formula 250 -- Fejer's Theorem 254 -- Complex Fourier Series 257 -- Part 3 Lebesgue Measure And Integration -- 16 Lebesgue Measure 263 -- Problem Of Measure 263 -- Lebesgue Outer Measure 268 -- Riemann Integrability 274 -- Measurable Sets 277 -- Structure Of Measurable Sets 283 -- A Nonmeasurable Set 289 -- Other Definitions 292 -- 17 Measurable Functions 296 -- Measurable Functions 296 -- Extended Real-valued Functions 302 -- Sequences Of Measurable Functions 304 -- Approximation Of Measurable Functions 306 -- 18 Lebesgue Integral 312 -- Simple Functions 312 -- Nonnegative Functions 314 -- General Case 322 -- Lebesgue's Dominated Convergence Theorem 328 -- Approximation Of Integrable Functions 333 -- 19 Additional Topics 337 -- Convergence In Measure 337 -- L[subscript P] Spaces 342 -- Approximation Of L[subscript P] Functions 350 -- More On Fourier Series 352 -- 20 Differentiation 359 -- Lebesgue's Differentiation Theorem 359 -- Absolute Continuity 370. N. L. Carothers. Includes Bibliographical References (p. 379-393) And Index. PART ONE. METRIC SPACES......Page 12 The Real Numbers......Page 14 Limits and Continuity......Page 25 Notes and Remarks......Page 28 Equivalence and Cardinality......Page 29 The Cantor Set......Page 36 Monotone Functions......Page 42 Notes and Remarks......Page 45 3 Metrics and Norms......Page 47 Metric Spaces......Page 48 Normed Vector Spaces......Page 50 More Inequalities......Page 54 Limits in Metric Spaces......Page 56 Notes and Remarks......Page 60 Open Sets......Page 62 Closed Sets......Page 64 The Relative Metric......Page 71 Notes and Remarks......Page 73 Continuous Functions......Page 74 Homeomorphisms......Page 80 The Space of Continuous Functions......Page 84 Notes and Remarks......Page 87 Connected Sets......Page 89 Notes and Remarks......Page 98 Totally Bounded Sets......Page 100 Complete Metric Spaces......Page 103 Fixed Points......Page 108 Completions......Page 113 Notes and Remarks......Page 117 8 Compactness......Page 119 Helly's First Theorem 2]0......Page Uniform Continuity......Page 125 Equivalent Metrics......Page 131 Notes and Remarks......Page 137 Discontinuous Functions......Page 139 The Baire Category Theorem......Page 142 Notes and Remarks......Page 147 PART TWO. FUNCTION SPACES......Page 148 Historical Background......Page 150 Pointwise and Uniform Convergence......Page 154 Interchanging Limits......Page 161 The Space of Bounded Functions......Page 164 Notes and Remarks......Page 171 The Weierstrass Theorem......Page 173 Trigonometric Polynomials......Page 181 Infinitely Differentiable Functions......Page 187 Equicontinuity......Page 189 Continuity and Category......Page 194 Notes and Remarks......Page 196 Algebras and Lattices......Page 199 The Stone-Weierstrass Theorem......Page 205 Notes and Remarks......Page 212 Functions of Bounded Variation......Page 213 Notes and Remarks......Page 223 Weights and Measures......Page 225 The Riemann-Stieltjes Integral......Page 226 The Space of Integrable Functions......Page 232 Integrators of Bounded Variation......Page 236 The Riemann Integral......Page 243 The Riesz Representation Theorem......Page 245 Other Definitions, Other Properties......Page 250 Notes and Remarks......Page 253 Preliminaries......Page 255 Dirichlet's Fonnula......Page 261 Fejer's Theorem......Page 265 Complex Fourier Series......Page 268 Notes and Remarks......Page 269 PART THREE. LEBESGUE MEASURE AND INTEGRATION......Page 272 The Problem of Measure......Page 274 Lebesgue Outer Measure......Page 279 Riemann Integrability......Page 285 Measurable Sets......Page 288 The Structure of Measurable Sets......Page 294 A Nonmeasurable Set......Page 300 Other Definitions......Page 303 Notes and Remarks......Page 304 Measurable Functions......Page 307 Extended Real-Valued Functions......Page 313 Sequences of Measurable Functions......Page 315 Approximation of Measurable Functions......Page 317 Notes and Remarks......Page 321 Simple Functions......Page 323 Nonnegative Functions......Page 325 The General Case......Page 333 Lebesgue's Dominated Convergence Theorem......Page 339 Approximation of Integrable Functions......Page 344 Notes and Remarks......Page 346 Convergence in Measure......Page 348 The Lp Spaces......Page 353 Approximation of Lp Functions......Page 361 More on Fourier Series......Page 363 Notes and Remarks......Page 367 Lebesgue's Differentiation Theorem......Page 370 Absolute Continuity......Page 381 Notes and Remarks......Page 388 References......Page 390 Symbol Index......Page 406 Topic Index......Page 408 This text for a course in Real Analysis addresses advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something of value to specialists and non-specialists alike. It considers three major topics: Metric and normed linear spaces, function spaces, and Lebesgue measure and integration on the line.Written in an informal, down-to-earth style, the book motivates the reader with an intuitive overview of new ideas, while still supplying full details and complete proofs. The author includes historical commentary with references to original works and alternate presentations, recommends expository and survey articles for non-specialists and technical articles for specialists, and provides a great many exercises and suggestions for further study.The author has written this text with the consideration of the heterogeneous audiences found at the masters level: students interested in pure and applied mathematics, statistics, education, engineering, and economics. This is a course in real analysis directed at advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something to specialists and non-specialists. The course consists of three major topics: metric and normed linear spaces, function spaces, and Lebesgue measure and integration on the line. In an informal style, the author gives motivation and overview of new ideas, while supplying full details and proofs. He includes historical commentary, recommends articles for specialists and non-specialists, and provides exercises and suggestions for further study. This text for a first graduate course in real analysis was written to accommodate the heterogeneous audiences found at the masters level: students interested in pure and applied mathematics, statistics, education, engineering, and economics. This course in real analysis is directed at advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something of value to specialists and nonspecialists alike. The text covers three major topics: metric and normed linear spaces, function spaces, and Lebesgue measure and integration on the line. In an informal, down-to-earth style, the author gives motivation and overview of new ideas, while still supplying full details and complete proofs. He provides a great many exercises and suggestions for further study.