Problems and Solutions in Real Analysis

Author: Masayoshi Hata

Publisher: World Scientific

ISBN: 981277601X

Category: Mathematics

Page: 292

View: 9187

This unique book provides a collection of more than 200 mathematical problems and their detailed solutions, which contain very useful tips and skills in real analysis. Each chapter has an introduction, in which some fundamental definitions and propositions are prepared. This also contains many brief historical comments on some significant mathematical results in real analysis together with useful references.Problems and Solutions in Real Analysis may be used as advanced exercises by undergraduate students during or after courses in calculus and linear algebra. It is also useful for graduate students who are interested in analytic number theory. Readers will also be able to completely grasp a simple and elementary proof of the prime number theorem through several exercises. The book is also suitable for non-experts who wish to understand mathematical analysis.
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Selected Problems in Real Analysis

Author: M. G. Goluzina,A. A. Lodkin,A. N. Podkorytov

Publisher: American Mathematical Soc.

ISBN: 9780821897386

Category: Mathematics

Page: 370

View: 5967

This book is intended for students wishing to deepen their knowledge of mathematical analysis and for those teaching courses in this area. It differs from other problem books in the greater difficulty of the problems, some of which are well-known theorems in analysis. Nonetheless, no special preparation is required to solve the majority of the problems. Brief but detailed solutions to most of the problems are given in the second part of the book. This book is unique in that the authors have aimed to systematize a range of problems that are found in sources that are almost inaccessible (especially to students) and in mathematical folklore.
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Problems in Real Analysis

Advanced Calculus on the Real Axis

Author: Teodora-Liliana Radulescu,Vicentiu D. Radulescu,Titu Andreescu

Publisher: Springer Science & Business Media

ISBN: 0387773797

Category: Mathematics

Page: 452

View: 2447

Problems in Real Analysis: Advanced Calculus on the Real Axis features a comprehensive collection of challenging problems in mathematical analysis that aim to promote creative, non-standard techniques for solving problems. This self-contained text offers a host of new mathematical tools and strategies which develop a connection between analysis and other mathematical disciplines, such as physics and engineering. A broad view of mathematics is presented throughout; the text is excellent for the classroom or self-study. It is intended for undergraduate and graduate students in mathematics, as well as for researchers engaged in the interplay between applied analysis, mathematical physics, and numerical analysis.
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Problems in Real Analysis

A Workbook with Solutions

Author: Charalambos D. Aliprantis,Owen Burkinshaw

Publisher: N.A

ISBN: 9780120502530

Category: Mathematics

Page: 403

View: 2653

This volume aims to teach the basic methods of proof and problem-solving by presenting the complete solutions to over 600 problems that appear in the companion "Principles of Real Analysis", 3rd edition.
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Selected Problems in Real Analysis

Author: B. M. Makarov

Publisher: American Mathematical Soc.

ISBN: 9780821809532

Category: Mathematics

Page: 370

View: 8422

This book is intended for students wishing to deepen their knowledge of mathematical analysis and for those teaching courses in this area. It differs from other problem books in the greater difficulty of the problems, some of which are well-known theorems in analysis. Nonetheless, no special preparation is required to solve the majority of the problems. Brief but detailed solutions to most of the problems are given in the second part of the book. This book is unique in that the authors have aimed to systematize a range of problems that are found in sources that are almost inaccessible (especially to students) and in mathematical folklore.
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A Problem Book in Real Analysis

Author: Asuman G. Aksoy,Mohamed A. Khamsi

Publisher: Springer Science & Business Media

ISBN: 1441912967

Category: Mathematics

Page: 254

View: 7007

Education is an admirable thing, but it is well to remember from time to time that nothing worth knowing can be taught. Oscar Wilde, “The Critic as Artist,” 1890. Analysis is a profound subject; it is neither easy to understand nor summarize. However, Real Analysis can be discovered by solving problems. This book aims to give independent students the opportunity to discover Real Analysis by themselves through problem solving. ThedepthandcomplexityofthetheoryofAnalysiscanbeappreciatedbytakingaglimpseatits developmental history. Although Analysis was conceived in the 17th century during the Scienti?c Revolution, it has taken nearly two hundred years to establish its theoretical basis. Kepler, Galileo, Descartes, Fermat, Newton and Leibniz were among those who contributed to its genesis. Deep conceptual changes in Analysis were brought about in the 19th century by Cauchy and Weierstrass. Furthermore, modern concepts such as open and closed sets were introduced in the 1900s. Today nearly every undergraduate mathematics program requires at least one semester of Real Analysis. Often, students consider this course to be the most challenging or even intimidating of all their mathematics major requirements. The primary goal of this book is to alleviate those concerns by systematically solving the problems related to the core concepts of most analysis courses. In doing so, we hope that learning analysis becomes less taxing and thereby more satisfying.
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Problems in Real and Complex Analysis

Author: Bernard R. Gelbaum

Publisher: Springer Science & Business Media

ISBN: 1461209250

Category: Mathematics

Page: 489

View: 2079

This text covers many principal topics in the theory of functions of a complex variable. These include, in real analysis, set algebra, measure and topology, real- and complex-valued functions, and topological vector spaces. In complex analysis, they include polynomials and power series, functions holomorphic in a region, entire functions, analytic continuation, singularities, harmonic functions, families of functions, and convexity theorems.
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Problems in Mathematical Analysis

Author: Biler

Publisher: CRC Press

ISBN: 9780824783129

Category: Mathematics

Page: 244

View: 5597

Chapter 1 poses 134 problems concerning real and complex numbers, chapter 2 poses 123 problems concerning sequences, and so it goes, until in chapter 9 one encounters 201 problems concerning functional analysis. The remainder of the book is given over to the presentation of hints, answers or referen
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Real Analysis and Probability

Solutions to Problems

Author: Robert P. Ash

Publisher: Academic Press

ISBN: 1483216187

Category: Mathematics

Page: 44

View: 8329

Real Analysis and Probability: Solutions to Problems presents solutions to problems in real analysis and probability. Topics covered range from measure and integration theory to functional analysis and basic concepts of probability; the interplay between measure theory and topology; conditional probability and expectation; the central limit theorem; and strong laws of large numbers in terms of martingale theory. Comprised of eight chapters, this volume begins with problems and solutions for the theory of measure and integration, followed by various applications of the basic integration theory. Subsequent chapters deal with functional analysis, paying particular attention to structures that can be defined on vector spaces; the connection between measure theory and topology; basic concepts of probability; and conditional probability and expectation. Strong laws of large numbers are also taken into account, first from the classical viewpoint, and then via martingale theory. The final chapter is devoted to the one-dimensional central limit problem, with emphasis on the fundamental role of Prokhorov's weak compactness theorem. This book is intended primarily for students taking a graduate course in probability.
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Advanced Real Analysis

Author: Anthony W. Knapp

Publisher: Springer Science & Business Media

ISBN: 9780817644420

Category: Mathematics

Page: 466

View: 5546

* Presents a comprehensive treatment with a global view of the subject * Rich in examples, problems with hints, and solutions, the book makes a welcome addition to the library of every mathematician
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Problems and Solutions for Undergraduate Real Analysis I

Author: Kit-Wing Yu

Publisher: 978-988-78797-5-6

ISBN: 9789887879756

Category:

Page: 212

View: 5422

The aim of Problems and Solutions for Undergraduate Real Analysis I, as the name reveals, is to assist undergraduate students or first-year students who study mathematics in learning their first rigorous real analysis course. The wide variety of problems, which are of varying difficulty, include the following topics: Elementary Set Algebra, the Real Number System, Countable and Uncountable Sets, Elementary Topology on Metric Spaces, Sequences in Metric Spaces, Series of Numbers, Limits and Continuity of Functions, Differentiation and the Riemann-Stieltjes Integral. Furthermore, the main features of this book are listed as follows: 1. The book contains 230 problems, which cover the topics mentioned above, with detailed and complete solutions. As a matter of fact, my solutions show every detail, every step and every theorem that I applied. 2. Each chapter starts with a brief and concise note of introducing the notations, terminologies, basic mathematical concepts or important/famous/frequently used theorems (without proofs) relevant to the topic. 3. Three levels of difficulty have been assigned to problems so that you can sharpen your mathematics step-by-step. 4. Different colors are used frequently in order to highlight or explain problems, examples, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) 5. An appendix about mathematical logic is included. It tells students what concepts of logic (e.g. techniques of proofs) are necessary in advanced mathematics.
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Problems and Proofs in Real Analysis

Theory of Measure and Integration

Author: J Yeh

Publisher: World Scientific Publishing Company

ISBN: 9814578525

Category: Mathematics

Page: 500

View: 1219

This volume consists of the proofs of 391 problems in Real Analysis: Theory of Measure and Integration (3rd Edition). Most of the problems in Real Analysis are not mere applications of theorems proved in the book but rather extensions of the proven theorems or related theorems. Proving these problems tests the depth of understanding of the theorems in the main text. This volume will be especially helpful to those who read Real Analysis in self-study and have no easy access to an instructor or an advisor.
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Problems and Proofs in Real Analysis

Theory of Measure and Integration

Author: James Yeh

Publisher: World Scientific Publishing Company Incorporated

ISBN: 9789814578509

Category: Mathematics

Page: 491

View: 6226

Companion volume to: Real analysis: theory of measure and integration (3rd ed.).
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Principles of Real Analysis

Author: Charalambos D. Aliprantis,Owen Burkinshaw

Publisher: Gulf Professional Publishing

ISBN: 9780120502578

Category: Mathematics

Page: 415

View: 8046

With the success of its previous editions, Principles of Real Analysis, Third Edition, continues to introduce students to the fundamentals of the theory of measure and functional analysis. In this thorough update, the authors have included a new chapter on Hilbert spaces as well as integrating over 150 new exercises throughout. The new edition covers the basic theory of integration in a clear, well-organized manner, using an imaginative and highly practical synthesis of the "Daniell Method" and the measure theoretic approach. Students will be challenged by the more than 600 exercises contained in the book. Topics are illustrated by many varied examples, and they provide clear connections between real analysis and functional analysis. Gives a unique presentation of integration theory Over 150 new exercises integrated throughout the text Presents a new chapter on Hilbert Spaces Provides a rigorous introduction to measure theory Illustrated with new and varied examples in each chapter Introduces topological ideas in a friendly manner Offers a clear connection between real analysis and functional analysis Includes brief biographies of mathematicians
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Problems in Real and Functional Analysis

Author: Alberto Torchinsky

Publisher: American Mathematical Soc.

ISBN: 1470420570

Category: Functional analysis

Page: 467

View: 3149

It is generally believed that solving problems is the most important part of the learning process in mathematics because it forces students to truly understand the definitions, comb through the theorems and proofs, and think at length about the mathematics. The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of conceptual problems (1,457 in total), ranging from easily accessible to thought provoking, mixing the practical and the theoretical aspects of the subject. Problems are grouped into ten chapters covering the main topics usually taught in courses on real and functional analysis. Each of these chapters opens with a brief reader's guide stating the needed definitions and basic results in the area and closes with a short description of the problems. - See more at: http://bookstore.ams.org/GSM-166/#sthash.ZMb1J6lg.dpuf It is generally believed that solving problems is the most important part of the learning process in mathematics because it forces students to truly understand the definitions, comb through the theorems and proofs, and think at length about the mathematics. The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of conceptual problems (1,457 in total), ranging from easily accessible to thought provoking, mixing the practical and the theoretical aspects of the subject. Problems are grouped into ten chapters covering the main topics usually taught in courses on real and functional analysis. Each of these chapters opens with a brief reader's guide stating the needed definitions and basic results in the area and closes with a short description of the problems. The Problem chapters are accompanied by Solution chapters, which include solutions to two-thirds of the problems. Students can expect the solutions to be written in a direct language that they can understand; usually the most "natural" rather than the most elegant solution is presented. The Problem chapters are accompanied by Solution chapters, which include solutions to two-thirds of the problems. Students can expect the solutions to be written in a direct language that they can understand; usually the most “natural” rather than the most elegant solution is presented. - See more at: http://bookstore.ams.org/GSM-166/#sthash.ZMb1J6lg.dpufhe Problem chapters are accompanied by Solution chapters, which include solutions to two-thirds of the - See more at: http://bookstore.ams.org/GSM-166/#sthash.ZMb1J6lg.dpuft is generally believed that solving problems is the most important part of the learning process in mathematics because it forces students to truly understand the definitions, comb through the theorems and proofs, and think at length about the mathematics. The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of - See more at: http://bookstore.ams.org/GSM-166/#sthash.ZMb1J6lg.dpufIt is generally believed that solving problems is the most important part of the learning process in mathematics because it forces students to truly understand the definitions, comb through the theorems and proofs, and think at length about the mathematics. The purpose of this book is to complement the existing literature in introductory real and functional analysis at the graduate level with a variety of conceptual problems (1,457 in total), ranging from easily accessible to thought provoking, mixing the practical and the theoretical aspects of the subject. Problems are grouped into ten chapters covering the main topics usually taught in courses on real and functional analysis. Each of these chapters opens with a brief reader's guide stating - See more at: http://bookstore.ams.org/GSM-166/#sthash.ZMb1J6lg.dpuf
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Funktionentheorie

Author: Eberhard Freitag,Rolf Busam

Publisher: Springer-Verlag

ISBN: 3662073501

Category: Mathematics

Page: 477

View: 6526

Die komplexen Zahlen haben ihre historischen Wurzeln im 16. Jahrhundert, sie entstanden bei dem Versuch, algebraische Gleichungen zu lösen. So führte schon G. CARDANO (1545) formale Ausdrücke wie zum Beispiel 5 ± V-15 ein, um Lösungen quadratischer und kubischer Gleichungen angeben zu können. R. BOMBELLI rechnete um 1560 bereits systematisch mit diesen Ausdrücken 3 und fand 4 als Lösung der Gleichung x = 15x + 4 in der verschlüsselten Form 4 = ~2 + V-121 + ~2 - V-121. Auch bei G. W. LEIBNIZ (1675) findet man Gleichungen dieser Art, wie z.B. J 1 + V-3 + J 1 - V-3 = v6. Im Jahre 1777 führte L. EULER die Bezeichnung i = yCI für die imaginäre Einheit ein. Der Fachausdruck "komplexe Zahl" stammt von C. F. GAUSS (1831). Die strenge Einführung der komplexen Zahlen als Paare reeller Zahlen geht auf W. R. HAMILTON (1837) zurück. Schon in der reellen Analysis ist es gelegentlich vorteilhaft, komplexe Zahlen einzuführen. Man denke beispielsweise an die Integration rationaler Funktio nen, die auf der Partialbruchentwicklung und damit auf dem Fundamentalsatz der Algebra beruht: Über dem Körper der komplexen Zahlen zerfällt jedes Polynom in ein Produkt von Linearfaktoren.
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Reelle und Komplexe Analysis

Author: Walter Rudin

Publisher: Walter de Gruyter

ISBN: 9783486591866

Category: Analysis - Lehrbuch

Page: 499

View: 686

Besonderen Wert legt Rudin darauf, dem Leser die Zusammenhänge unterschiedlicher Bereiche der Analysis zu vermitteln und so die Grundlage für ein umfassenderes Verständnis zu schaffen. Das Werk zeichnet sich durch seine wissenschaftliche Prägnanz und Genauigkeit aus und hat damit die Entwicklung der modernen Analysis in nachhaltiger Art und Weise beeinflusst. Der "Baby-Rudin" gehört weltweit zu den beliebtesten Lehrbüchern der Analysis und ist in 13 Sprachen übersetzt. 1993 wurde es mit dem renommierten Steele Prize for Mathematical Exposition der American Mathematical Society ausgezeichnet. Übersetzt von Uwe Krieg.
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