Basic Real Analysis
Autor Houshang H. Sohraben Limba Engleză Paperback – 28 sep 2012
În cadrul curriculumului de matematică pură și aplicată, analiza reală reprezintă pilonul fundamental pe care se sprijină înțelegerea riguroasă a calculului diferențial și integral. Basic Real Analysis de Houshang H. Sohrab se poziționează ca un text de tranziție esențial, facilitând trecerea de la calculul computațional la demonstrația formală. Găsim în această lucrare o abordare care prioritizează rigoarea, fără a sacrifica însă accesibilitatea pentru cititorul aflat la prima interacțiune cu limbajul matematic avansat.
Notăm cu interes structura progresivă a volumului, care debutează cu bazele teoriei mulțimilor și cardinalitatea numerelor, trecând prin studiul secvențelor și seriilor, pentru a ajunge la topologia spațiilor metrice. Această ediție extinde cadrul propus de Analysis de Richard Beals prin includerea unor versiuni mai detaliate ale teoremelor fundamentale și prin rescrierea completă a secțiunilor despre măsura Lebesgue. Dacă Analysis se concentrează pe o perspectivă largă ce include ecuațiile diferențiale, volumul de față alege să aprofundeze analiza reală pură, oferind demonstrații optimizate pentru a fi parcurse facil.
Elementul distinctiv al cărții publicate de Birkhäuser Boston este densitatea aplicațiilor practice: cele peste 650 de exerciții integrate și problemele de la finalul capitolelor transformă textul dintr-o expunere teoretică într-un instrument de lucru activ. Față de alte titluri clasice menționate în recenziile de specialitate, precum cele ale lui Rudin sau Lang, abordarea lui Sohrab este mai puțin abruptă, oferind context motivațional și exemple lucrate pas cu pas care ajută la fixarea conceptelor de continuitate uniformă sau aproximare funcțională.
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Specificații
ISBN-13: 9781461265030
ISBN-10: 1461265037
Pagini: 576
Ilustrații: XIII, 559 p.
Dimensiuni: 155 x 235 x 30 mm
Greutate: 0.79 kg
Ediția:Softcover reprint of the original 1st ed. 2003
Editura: Birkhäuser Boston
Colecția Birkhäuser
Locul publicării:Boston, MA, United States
ISBN-10: 1461265037
Pagini: 576
Ilustrații: XIII, 559 p.
Dimensiuni: 155 x 235 x 30 mm
Greutate: 0.79 kg
Ediția:Softcover reprint of the original 1st ed. 2003
Editura: Birkhäuser Boston
Colecția Birkhäuser
Locul publicării:Boston, MA, United States
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Public țintă
GraduateDe ce să citești această carte
Această carte este recomandată studenților care doresc să stăpânească fundamentele analizei reale printr-un parcurs riguros, dar ghidat. Cititorul câștigă acces la o bază solidă de probleme rezolvate și la o expunere clară a integralei Lebesgue, esențială pentru cercetarea avansată. Este o resursă valoroasă pentru auto-studiu, datorită apendicelor tehnice și a stilului pedagogic care demistifică demonstrațiile complexe.
Despre autor
Houshang H. Sohrab este un matematician cu o vastă experiență academică, recunoscut pentru contribuțiile sale pedagogice în domeniul analizei matematice. Lucrarea sa, publicată sub egida Birkhäuser, reflectă o înțelegere profundă a dificultăților pe care studenții le întâmpină la nivel postuniversitar. Autorul pune accent pe claritatea expunerii și pe selecția judicioasă a referințelor bibliografice, transformând Basic Real Analysis într-un titlu de referință în bibliografia de specialitate pentru cursurile de analiză reală și teoria măsurii.
Descriere scurtă
One of the bedrocks of any mathematics education, the study of real analysis introduces students both to mathematical rigor and to the deep theorems and counterexamples that arise from such rigor: for instance, the construction of number systems, the Cantor Set, the Weierstrass nowhere differentiable function, and the Weierstrass approximation theorem. Basic Real Analysis is a modern, systematic text that presents the fundamentals and touchstone results of the subject in full rigor, but in a style that requires little prior familiarity with proofs or mathematical language.
Key features include:
* A broad view of mathematics throughout the book
* Treatment of all concepts for real numbers first, with extensions to metric spaces later, in a separate chapter
* Elegant proofs
* Excellent choice of topics
* Numerous examples and exercises to enforce methodology; exercises integrated into the main text, as well as at the end of each chapter
* Emphasis on monotone functions throughout
* Good development of integration theory
* Special topics on Banach and Hilbert spaces and Fourier series, often not included in many courses on real analysis
* Solid preparation for deeper study of functional analysis
* Chapter on elementary probability
* Comprehensive bibliography and index
* Solutions manual available to instructors upon request
By covering all the basics and developing rigor simultaneously, this introduction to real analysis is ideal for senior undergraduates and beginning graduate students, both as a classroom text or for self-study. With its wide range of topics and its view of real analysis in a larger context, the book will be appropriate for more advanced readers as well.
Key features include:
* A broad view of mathematics throughout the book
* Treatment of all concepts for real numbers first, with extensions to metric spaces later, in a separate chapter
* Elegant proofs
* Excellent choice of topics
* Numerous examples and exercises to enforce methodology; exercises integrated into the main text, as well as at the end of each chapter
* Emphasis on monotone functions throughout
* Good development of integration theory
* Special topics on Banach and Hilbert spaces and Fourier series, often not included in many courses on real analysis
* Solid preparation for deeper study of functional analysis
* Chapter on elementary probability
* Comprehensive bibliography and index
* Solutions manual available to instructors upon request
By covering all the basics and developing rigor simultaneously, this introduction to real analysis is ideal for senior undergraduates and beginning graduate students, both as a classroom text or for self-study. With its wide range of topics and its view of real analysis in a larger context, the book will be appropriate for more advanced readers as well.
Cuprins
1 Set Theory.- 1.1 Rings and Algebras of Sets.- 1.2 Relations and Functions.- 1.3 Basic Algebra, Counting, and Arithmetic.- 1.4 Infinite Direct Products, Axiom of Choice, and Cardinal Numbers.- 1.5 Problems.- 2 Sequences and Series of Real Numbers.- 2.1 Real Numbers.- 2.2 Sequences in ?.- 2.3 Infinite Series.- 2.4 Unordered Series and Summability.- 2.5 Problems.- 3 Limits of Functions.- 3.1 Bounded and Monotone Functions.- 3.2 Limits of Functions.- 3.3 Properties of Limits.- 3.4 One-sided Limits and Limits Involving Infinity.- 3.5 Indeterminate Forms, Equivalence, Landau’s Little “oh” and Big “Oh”.- 3.6 Problems.- 4 Topology of ? and Continuity.- 4.1 Compact and Connected Subsets of ?.- 4.2 The Cantor Set.- 4.3 Continuous Functions.- 4.4 One-sided Continuity, Discontinuity, and Monotonicity.- 4.5 Extreme Value and Intermediate Value Theorems.- 4.6 Uniform Continuity.- 4.7 Approximation by Step, Piecewise Linear, and Polynomial Functions.- 4.8 Problems.- 5 Metric Spaces.- 5.1 Metrics and Metric Spaces.- 5.2 Topology of a Metric Space.- 5.3 Limits, Cauchy Sequences, and Completeness.- 5.4 Continuity.- 5.5 Uniform Continuity and Continuous Extensions.- 5.6 Compact Metric Spaces.- 5.7 Connected Metric Spaces.- 5.8 Problems.- 6 The Derivative.- 6.1 Differentiability.- 6.2 Derivatives of Elementary Functions.- 6.3 The Differential Calculus.- 6.4 Mean Value Theorems.- 6.5 L’Hôpital’s Rule.- 6.6 Higher Derivatives and Taylor’s Formula.- 6.7 Convex Functions.- 6.8 Problems.- 7 The Riemann Integral.- 7.1 Tagged Partitions and Riemann Sums.- 7.2 Some Classes of Integrable Functions.- 7.3 Sets of Measure Zero and Lebesgue’s Integrability Criterion.- 7.4 Properties of the Riemann Integral.- 7.5 Fundamental Theorem of Calculus.- 7.6 Functions of BoundedVariation.- 7.7 Problems.- 8 Sequences and Series of Functions.- 8.1 Complex Numbers.- 8.2 Pointwise and Uniform Convergence.- 8.3 Uniform Convergence and Limit Theorems.- 8.4 Power Series.- 8.5 Elementary Transcendental Functions.- 8.6 Fourier Series.- 8.7 Problems.- 9 Normed and Function Spaces.- 9.1 Norms and Normed Spaces.- 9.2 Banach Spaces.- 9.3 Hilbert Spaces.- 9.4 Function Spaces.- 9.5 Problems.- 10 The Lebesgue Integral (F. Riesz’s Approach).- 10.1 Improper Riemann Integrals.- 10.2 Step Functions and Their Integrals.- 10.3 Convergence Almost Everywhere.- 10.4 The Lebesgue Integral.- 10.5 Convergence Theorems.- 10.6 The Banach Space L1.- 10.7 Problems.- 11 Lebesgue Measure.- 11.1 Measurable Functions.- 11.2 Measurable Sets and Lebesgue Measure.- 11.3 Measurability (Lebesgue’s Definition).- 11.4 The Theorems of Egorov, Lusin, and Steinhaus.- 11.5 Regularity of Lebesgue Measure.- 11.6 Lebesgue’s Outer and Inner Measures.- 11.7 The Hilbert Spaces L2(E, % MathType!MTEF!2!1!+-% feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9% gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav% P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC% 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq% aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe% qaamaaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv% 39gaiyaacqWFfcVraaa!47BC!$$\mathbb{F}$$).- 11.8 Problems.- 12 General Measure and Probability.- 12.1 Measures and Measure Spaces.- 12.2 Measurable Functions.- 12.3 Integration.- 12.4 Probability.- 12.5 Problems.- A Construction of Real Numbers.- References.
Recenzii
"Students who find Goffman's ‘Real Functions’ (1953), Halmos's ‘Measure Theory’ (1950), Hewitt and Stromberg's ‘Real and Abstract Analysis’ (1965), Lang's (1969) or Royden's ‘Real Analysis’ (1963), or Rudin's (1973) or Yosida's ‘Functional Analysis’ (1965) to be too hard, or too easy, may find Sohrab's presentation just right. Problems and exercises abound; an appendix constructs the reals as the Cauchy (sequential) completion of the rationals; references are copious and judiciously chosen; and a detailed index brings up the rear. . . . Recommended."
—CHOICE
"This book is intended as a text for a one-year course for senior undergraduates or beginning graduate students, though it seems to the reviewer that it contains more than enough material for one year's study. . . . The quality of the exposition is good: strong and complete versions of theorems are preferred, and the material is organised so that all the proofs are of easily manageable length; motivational comments are helpful, and there are plenty of illustrative examples. The reader is strongly encouraged to learn by doing: exercises are sprinkled liberally throughout the text and each chapter ends with a set of problems, about 650 in all, some of which are of considerable intrinsic interest."
—MATHEMATICAL REVIEWS
"The book is a clear and well structured introduction to real analysis aimed at senior undergraduate and beginning graduate students. . . . The author managed to confine within a reasonable size book, all the basic concepts in real analysis and also some developed topics . . . The text contains carefully worked out examples which contribute motivating and helping to understand the theory. There is also an excellent selection of exercises within the text and problem sections at the end of each chapter. In fact this textbook can serve as a source of examples and exercises in real analysis. . . . This book can behighly recommended as a good reference on real analysis."
—ZENTRALBLATT MATH
—CHOICE
"This book is intended as a text for a one-year course for senior undergraduates or beginning graduate students, though it seems to the reviewer that it contains more than enough material for one year's study. . . . The quality of the exposition is good: strong and complete versions of theorems are preferred, and the material is organised so that all the proofs are of easily manageable length; motivational comments are helpful, and there are plenty of illustrative examples. The reader is strongly encouraged to learn by doing: exercises are sprinkled liberally throughout the text and each chapter ends with a set of problems, about 650 in all, some of which are of considerable intrinsic interest."
—MATHEMATICAL REVIEWS
"The book is a clear and well structured introduction to real analysis aimed at senior undergraduate and beginning graduate students. . . . The author managed to confine within a reasonable size book, all the basic concepts in real analysis and also some developed topics . . . The text contains carefully worked out examples which contribute motivating and helping to understand the theory. There is also an excellent selection of exercises within the text and problem sections at the end of each chapter. In fact this textbook can serve as a source of examples and exercises in real analysis. . . . This book can behighly recommended as a good reference on real analysis."
—ZENTRALBLATT MATH
Caracteristici
Request lecturer material: sn.pub/lecturer-material
Notă biografică
Houshang H. Sohrab is a Professor of Mathematics at Towson University.
Textul de pe ultima copertă
This expanded second edition presents the fundamentals and touchstone results of real analysis in full rigor, but in a style that requires little prior familiarity with proofs or mathematical language.
The text is a comprehensive and largely self-contained introduction to the theory of real-valued functions of a real variable. The chapters on Lebesgue measure and integral have been rewritten entirely and greatly improved. They now contain Lebesgue’s differentiation theorem as well as his versions of the Fundamental Theorem(s) of Calculus.
With expanded chapters, additional problems, and an expansive solutions manual, Basic Real Analysis, Second Edition, is ideal for senior undergraduates and first-year graduate students, both as a classroom text and a self-study guide.
Reviews of first edition:
The book is a clear and well-structured introduction to real analysis aimed at senior undergraduate and beginning graduate students. The prerequisites are few, but a certain mathematical sophistication is required. ... The text contains carefully worked out examples which contribute motivating and helping to understand the theory. There is also an excellent selection of exercises within the text and problem sections at the end of each chapter. In fact, this textbook can serve as a source of examples and exercises in real analysis.
—Zentralblatt MATH
The quality of the exposition is good: strong and complete versions of theorems are preferred, and the material is organised so that all the proofs are of easily manageable length; motivational comments are helpful, and there are plenty of illustrative examples. The reader is strongly encouraged to learn by doing: exercises are sprinkled liberally throughout the text and each chapter ends with a set of problems, about 650 in all, some of which are of considerable intrinsic interest.
—Mathematical Reviews
[This text] introduces upper-division undergraduate or first-year graduate students to real analysis.... Problems and exercises abound; an appendix constructs the reals as the Cauchy (sequential) completion of the rationals; references are copious and judiciously chosen; and a detailed index brings up the rear.
—CHOICE Reviews
The text is a comprehensive and largely self-contained introduction to the theory of real-valued functions of a real variable. The chapters on Lebesgue measure and integral have been rewritten entirely and greatly improved. They now contain Lebesgue’s differentiation theorem as well as his versions of the Fundamental Theorem(s) of Calculus.
With expanded chapters, additional problems, and an expansive solutions manual, Basic Real Analysis, Second Edition, is ideal for senior undergraduates and first-year graduate students, both as a classroom text and a self-study guide.
Reviews of first edition:
The book is a clear and well-structured introduction to real analysis aimed at senior undergraduate and beginning graduate students. The prerequisites are few, but a certain mathematical sophistication is required. ... The text contains carefully worked out examples which contribute motivating and helping to understand the theory. There is also an excellent selection of exercises within the text and problem sections at the end of each chapter. In fact, this textbook can serve as a source of examples and exercises in real analysis.
—Zentralblatt MATH
The quality of the exposition is good: strong and complete versions of theorems are preferred, and the material is organised so that all the proofs are of easily manageable length; motivational comments are helpful, and there are plenty of illustrative examples. The reader is strongly encouraged to learn by doing: exercises are sprinkled liberally throughout the text and each chapter ends with a set of problems, about 650 in all, some of which are of considerable intrinsic interest.
—Mathematical Reviews
[This text] introduces upper-division undergraduate or first-year graduate students to real analysis.... Problems and exercises abound; an appendix constructs the reals as the Cauchy (sequential) completion of the rationals; references are copious and judiciously chosen; and a detailed index brings up the rear.
—CHOICE Reviews