The Real Numbers and Real Analysis

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Author: Ethan D. Bloch

Publisher: Springer Science & Business Media

ISBN: 0387721762

Category: Mathematics

Page: 554

View: 7371

This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. It is organized in a distinctive, flexible way that would make it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus. The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.

The Real Numbers

An Introduction to Set Theory and Analysis

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Author: John Stillwell

Publisher: Springer Science & Business Media

ISBN: 331901577X

Category: Mathematics

Page: 244

View: 8900

While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.

A First Course in Real Analysis

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Author: Sterling K. Berberian

Publisher: Springer Science & Business Media

ISBN: 1441985484

Category: Mathematics

Page: 240

View: 8682

Mathematics is the music of science, and real analysis is the Bach of mathematics. There are many other foolish things I could say about the subject of this book, but the foregoing will give the reader an idea of where my heart lies. The present book was written to support a first course in real analysis, normally taken after a year of elementary calculus. Real analysis is, roughly speaking, the modern setting for Calculus, "real" alluding to the field of real numbers that underlies it all. At center stage are functions, defined and taking values in sets of real numbers or in sets (the plane, 3-space, etc.) readily derived from the real numbers; a first course in real analysis traditionally places the emphasis on real-valued functions defined on sets of real numbers. The agenda for the course: (1) start with the axioms for the field ofreal numbers, (2) build, in one semester and with appropriate rigor, the foun dations of calculus (including the "Fundamental Theorem"), and, along the way, (3) develop those skills and attitudes that enable us to continue learning mathematics on our own. Three decades of experience with the exercise have not diminished my astonishment that it can be done.

An Introduction to Real Analysis

The Commonwealth and International Library: Mathematical Topics

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Author: Derek G. Ball

Publisher: Elsevier

ISBN: 1483158969

Category: Mathematics

Page: 324

View: 2400

An Introduction to Real Analysis presents the concepts of real analysis and highlights the problems which necessitate the introduction of these concepts. Topics range from sets, relations, and functions to numbers, sequences, series, derivatives, and the Riemann integral. This volume begins with an introduction to some of the problems which are met in the use of numbers for measuring, and which provide motivation for the creation of real analysis. Attention then turns to real numbers that are built up from natural numbers, with emphasis on integers, rationals, and irrationals. The chapters that follow explore the conditions under which sequences have limits and derive the limits of many important sequences, along with functions of a real variable, Rolle's theorem and the nature of the derivative, and the theory of infinite series and how the concepts may be applied to decimal representation. The book also discusses some important functions and expansions before concluding with a chapter on the Riemann integral and the problem of area and its measurement. Throughout the text the stress has been upon concepts and interesting results rather than upon techniques. Each chapter contains exercises meant to facilitate understanding of the subject matter. This book is intended for students in colleges of education and others with similar needs.

Elements of Real Analysis

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Author: David A. Sprecher

Publisher: Courier Corporation

ISBN: 0486153258

Category: Mathematics

Page: 368

View: 3479

Classic text explores intermediate steps between basics of calculus and ultimate stage of mathematics — abstraction and generalization. Covers fundamental concepts, real number system, point sets, functions of a real variable, Fourier series, more. Over 500 exercises.

A Course in Mathematical Analysis: Volume 1, Foundations and Elementary Real Analysis

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Author: D. J. H. Garling

Publisher: Cambridge University Press

ISBN: 1107311381

Category: Mathematics

Page: N.A

View: 499

The three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in their first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and instructors. This first volume focuses on the analysis of real-valued functions of a real variable. Besides developing the basic theory it describes many applications, including a chapter on Fourier series. It also includes a Prologue in which the author introduces the axioms of set theory and uses them to construct the real number system. Volume 2 goes on to consider metric and topological spaces and functions of several variables. Volume 3 covers complex analysis and the theory of measure and integration.

Real Analysis

A Constructive Approach

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Author: Mark Bridger

Publisher: John Wiley & Sons

ISBN: 1118031563

Category: Mathematics

Page: 320

View: 4801

A unique approach to analysis that lets you apply mathematics across a range of subjects This innovative text sets forth a thoroughly rigorous modern account of the theoretical underpinnings of calculus: continuity, differentiability, and convergence. Using a constructive approach, every proof of every result is direct and ultimately computationally verifiable. In particular, existence is never established by showing that the assumption of non-existence leads to a contradiction. The ultimate consequence of this method is that it makes sense—not just to math majors but also to students from all branches of the sciences. The text begins with a construction of the real numbers beginning with the rationals, using interval arithmetic. This introduces readers to the reasoning and proof-writing skills necessary for doing and communicating mathematics, and it sets the foundation for the rest of the text, which includes: Early use of the Completeness Theorem to prove a helpful Inverse Function Theorem Sequences, limits and series, and the careful derivation of formulas and estimates for important functions Emphasis on uniform continuity and its consequences, such as boundedness and the extension of uniformly continuous functions from dense subsets Construction of the Riemann integral for functions uniformly continuous on an interval, and its extension to improper integrals Differentiation, emphasizing the derivative as a function rather than a pointwise limit Properties of sequences and series of continuous and differentiable functions Fourier series and an introduction to more advanced ideas in functional analysis Examples throughout the text demonstrate the application of new concepts. Readers can test their own skills with problems and projects ranging in difficulty from basic to challenging. This book is designed mainly for an undergraduate course, and the author understands that many readers will not go on to more advanced pure mathematics. He therefore emphasizes an approach to mathematical analysis that can be applied across a range of subjects in engineering and the sciences.

Analysis I

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Author: Wolfgang Walter

Publisher: Springer-Verlag

ISBN: 3662057085

Category: Mathematics

Page: 388

View: 7411

Das vorliegende Buch ist der erste Band eines zweibändigen Werkes über Analysis und behandelt die Funktionen einer reellen Veränderlichen. In der komplexen Analysis beschränkt es sich im wesentlichen auf Potenzreihen. Es enthält insbesondere den Stoff, welcher üblicherweise im ersten Semester einer einführenden Analysis-Vorlesung für Mathematiker, Physiker und Informatiker geboten wird, und geht an einigen Stellen darüber hinaus. Das Buch wendet sich an Studenten, denen es sich als ein hilfreicher Begleiter der Vorlesung und eine Quelle zur Vertiefung des Gegenstandes anbietet, an die im Beruf stehen den Mathematiker, besonders an die Lehrer an weiterführenden Schulen, und schließlich an alle, die etwas über die Analysis und ihre Bedeutung im größeren naturwissenschaftlichen und kulturellen Zusammenhang erfahren möchten. Damit sind wir bei einem wesentlichen Anliegen der Lehrbuchreihe "Grundwissen Mathematik", dem historischen Bezug. Die mathematischen Be griffe und Inhalte der Analysis sind nicht vom Himmel der reinen Erkenntnis gefallen, und kein Denker im Elfenbeinturm hat sie ersonnen. Die europäische Geistesgeschichte beginnt dort, wo Natur nicht mehr als rätselhaftes, von un heimlichen höheren Mächten gesteuertes Geschehen, sondern als rational erklärbar verstanden wird: bei den jonischen Philosophen des 6. vorchrist lichen Jahrhunderts. Die Analysis ist entstanden in der Verfolgung dieses Zie les, die Welt rational zu durchdringen und ihre Gesetzmäßigkeiten zu finden. Ihre Geschichte ist ein Stück Kulturgeschichte.

Introduction to Real Analysis

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Author: Michael J. Schramm

Publisher: Courier Corporation

ISBN: 0486131920

Category: Mathematics

Page: 384

View: 2041

This text forms a bridge between courses in calculus and real analysis. Suitable for advanced undergraduates and graduate students, it focuses on the construction of mathematical proofs. 1996 edition.

Elementary Real Analysis

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Author: Brian S. Thomson,Judith B. Bruckner,Andrew M. Bruckner

Publisher: ClassicalRealAnalysis.com

ISBN: 1434841618

Category: Mathematics

Page: 365

View: 3137

This is the second edition of the title originally published by Prentice Hall (Pearson) in 2001. Here is the reference information for the first edition:[TBB] Elementary Real Analysis, Brian S. Thomson, Judith B. Bruckner,Andrew M. Bruckner. Prentice-Hall, 2001, xv 735 pp. [ISBN 0-13-019075-61]The present title contains Chapters 1-8. The full version containing all of the chapters is also available as a trade paperback. A hypertexted PDF file of the entire text is available free for download on www.classicalrealanalysis.com.Chapter 1. Real NumbersChapter 2. SequencesChapter 3. Infinite sumsChapter 4. Sets of real numbersChapter 5. Continuous functionsChapter 6. More on continuous functions and setsChapter 7. DifferentiationChapter 8. The integral

Russell's Unknown Logicism

A Study in the History and Philosophy of Mathematics

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Author: S. Gandon

Publisher: Springer

ISBN: 1137024658

Category: Mathematics

Page: 263

View: 996

In this excellent book Sebastien Gandon focuses mainly on Russell's two major texts, Principa Mathematica and Principle of Mathematics , meticulously unpicking the details of these texts and bringing a new interpretation of both the mathematical and the philosophical content. Winner of The Bertrand Russell Society Book Award 2013.

The Foundations of Analysis: A Straightforward Introduction

Book 1 Logic, Sets and Numbers

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Author: K. G. Binmore

Publisher: CUP Archive

ISBN: 9780521299152

Category: Mathematics

Page: 144

View: 9806

This book is entirely self-contained but, it will be of most use to university or college students who are taking, or who have taken, an introductory course in analysis. There are a large number of examples and exercises and, where necessary, hints to the solutions are provided.

Computable Analysis

An Introduction

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Author: Klaus Weihrauch

Publisher: Springer Science & Business Media

ISBN: 9783540668176

Category: Computers

Page: 288

View: 7097

Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis. It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics. Many examples illustrate the new concepts while numerous exercises of varying difficulty extend the material and stimulate readers to work actively on the text.

Real Analysis with an Introduction to Wavelets and Applications

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Author: Don Hong,Jianzhong Wang,Robert Gardner

Publisher: Elsevier

ISBN: 9780080540313

Category: Mathematics

Page: 392

View: 5194

Real Analysis with an Introduction to Wavelets and Applications is an in-depth look at real analysis and its applications, including an introduction to wavelet analysis, a popular topic in "applied real analysis". This text makes a very natural connection between the classic pure analysis and the applied topics, including measure theory, Lebesgue Integral, harmonic analysis and wavelet theory with many associated applications. The text is relatively elementary at the start, but the level of difficulty steadily increases The book contains many clear, detailed examples, case studies and exercises Many real world applications relating to measure theory and pure analysis Introduction to wavelet analysis

Optimization

Insights and Applications

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Author: Jan Brinkhuis,Vladimir Tikhomirov

Publisher: Princeton University Press

ISBN: 9781400829361

Category: Mathematics

Page: 680

View: 6004

This self-contained textbook is an informal introduction to optimization through the use of numerous illustrations and applications. The focus is on analytically solving optimization problems with a finite number of continuous variables. In addition, the authors provide introductions to classical and modern numerical methods of optimization and to dynamic optimization. The book's overarching point is that most problems may be solved by the direct application of the theorems of Fermat, Lagrange, and Weierstrass. The authors show how the intuition for each of the theoretical results can be supported by simple geometric figures. They include numerous applications through the use of varied classical and practical problems. Even experts may find some of these applications truly surprising. A basic mathematical knowledge is sufficient to understand the topics covered in this book. More advanced readers, even experts, will be surprised to see how all main results can be grounded on the Fermat-Lagrange theorem. The book can be used for courses on continuous optimization, from introductory to advanced, for any field for which optimization is relevant.

Introduction to Analysis

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Author: Maxwell Rosenlicht

Publisher: Courier Corporation

ISBN: 0486134687

Category: Mathematics

Page: 272

View: 9721

Written for junior and senior undergraduates, this remarkably clear and accessible treatment covers set theory, the real number system, metric spaces, continuous functions, Riemann integration, multiple integrals, and more. 1968 edition.

Real and Abstract Analysis

A modern treatment of the theory of functions of a real variable

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Author: Edwin Hewitt,Karl Stromberg

Publisher: Springer-Verlag

ISBN: 3662297949

Category: Mathematics

Page: 476

View: 8460

The Mathematical Education of Teachers II

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Author: Conference Board of the Mathematical Sciences

Publisher: American Mathematical Soc.

ISBN: 0821869264

Category: Mathematics

Page: 86

View: 517

This report is a resource for those who teach mathematics and statistics to pre-K-12 mathematics teachers, both future teachers and those who already teach in our nation's schools. The report makes recommendations for the mathematics that teachers should know and how they should come to know that mathematics.