Preview this item
Preview this item
Checking...

# Basic Real Analysis

Author: Houshang H Sohrab Boston, MA : Birkhäuser Boston : Imprint : Birkhäuser, 2003. eBook : Document : EnglishView all editions and formats 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.  Read more... (not yet rated) 0 with reviews - Be the first.

## Find a copy in the library

Finding libraries that hold this item...

## Details

Genre/Form: Electronic books Print version: Document, Internet resource Internet Resource, Computer File Houshang H Sohrab Find more information about: Houshang H Sohrab 9780817682323 0817682325 853264095 1 online resource (xiii, 559 pages) 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 Bounded Variation -- 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. by Houshang H. Sohrab.

### Abstract:

Basic Real Analysis demonstrates the richness of real analysis, giving students an introduction both to mathematical rigor and to the deep theorems and counter examples that arise from such rigor.  Read more...

## Reviews

Editorial reviews

Publisher Synopsis

"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), Read more...

User-contributed reviews

Be the first.

## Similar Items

### Related Subjects:(3)

Confirm this request

You may have already requested this item. Please select Ok if you would like to proceed with this request anyway.

### Primary Entity

<http://www.worldcat.org/oclc/853264095> # Basic Real Analysis
a schema:MediaObject, schema:Book, schema:CreativeWork ;
library:oclcnum "853264095" ;
library:placeOfPublication <http://experiment.worldcat.org/entity/work/data/709949#Place/boston_ma> ; # Boston, MA
library:placeOfPublication <http://id.loc.gov/vocabulary/countries/mau> ;
schema:about <http://id.worldcat.org/fast/943472> ; # Global analysis (Mathematics)
schema:about <http://id.worldcat.org/fast/1002068> ; # Logic, Symbolic and mathematical
schema:bookFormat schema:EBook ;
schema:creator <http://viaf.org/viaf/8314558> ; # Houshang H. Sohrab
schema:datePublished "2003" ;
schema:description "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 Bounded Variation -- 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."@en ;
schema:description "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."@en ;
schema:exampleOfWork <http://worldcat.org/entity/work/id/709949> ;
schema:genre "Electronic books"@en ;
schema:inLanguage "en" ;
schema:isSimilarTo <http://worldcat.org/entity/work/data/709949#CreativeWork/> ;
schema:name "Basic Real Analysis"@en ;
schema:productID "853264095" ;
schema:publication <http://www.worldcat.org/title/-/oclc/853264095#PublicationEvent/boston_ma_birkhauser_boston_imprint_birkhauser_2003> ;
schema:publisher <http://experiment.worldcat.org/entity/work/data/709949#Agent/birkhauser_boston> ; # Birkhäuser Boston
schema:publisher <http://experiment.worldcat.org/entity/work/data/709949#Agent/birkhauser> ; # Birkhäuser
schema:publisher <http://experiment.worldcat.org/entity/work/data/709949#Agent/imprint> ; # Imprint
schema:url <http://dx.doi.org/10.1007/978-0-8176-8232-3> ;
schema:workExample <http://dx.doi.org/10.1007/978-0-8176-8232-3> ;
schema:workExample <http://worldcat.org/isbn/9780817682323> ;
wdrs:describedby <http://www.worldcat.org/title/-/oclc/853264095> ;
.

### Related Entities

<http://experiment.worldcat.org/entity/work/data/709949#Agent/birkhauser_boston> # Birkhäuser Boston
a bgn:Agent ;
schema:name "Birkhäuser Boston" ;
.

<http://id.worldcat.org/fast/1002068> # Logic, Symbolic and mathematical
a schema:Intangible ;
schema:name "Logic, Symbolic and mathematical"@en ;
.

<http://id.worldcat.org/fast/1012163> # Mathematics
a schema:Intangible ;
schema:name "Mathematics"@en ;
.

<http://id.worldcat.org/fast/943472> # Global analysis (Mathematics)
a schema:Intangible ;
schema:name "Global analysis (Mathematics)"@en ;
.

<http://viaf.org/viaf/8314558> # Houshang H. Sohrab
a schema:Person ;
schema:familyName "Sohrab" ;
schema:givenName "Houshang H." ;
schema:name "Houshang H. Sohrab" ;
.

<http://worldcat.org/isbn/9780817682323>
a schema:ProductModel ;
schema:isbn "0817682325" ;
schema:isbn "9780817682323" ;
.

Content-negotiable representations