Measure and Integral

Author: Richard L. Wheeden
Publisher: CRC Press
ISBN: 9781498702904
Release Date: 2015-04-24
Genre: Mathematics

Now considered a classic text on the topic, Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis by first developing the theory of measure and integration in the simple setting of Euclidean space, and then presenting a more general treatment based on abstract notions characterized by axioms and with less geometric content. Published nearly forty years after the first edition, this long-awaited Second Edition also: Studies the Fourier transform of functions in the spaces L1, L2, and Lp, 1 p Shows the Hilbert transform to be a bounded operator on L2, as an application of the L2 theory of the Fourier transform in the one-dimensional case Covers fractional integration and some topics related to mean oscillation properties of functions, such as the classes of Hölder continuous functions and the space of functions of bounded mean oscillation Derives a subrepresentation formula, which in higher dimensions plays a role roughly similar to the one played by the fundamental theorem of calculus in one dimension Extends the subrepresentation formula derived for smooth functions to functions with a weak gradient Applies the norm estimates derived for fractional integral operators to obtain local and global first-order Poincaré–Sobolev inequalities, including endpoint cases Proves the existence of a tangent plane to the graph of a Lipschitz function of several variables Includes many new exercises not present in the first edition This widely used and highly respected text for upper-division undergraduate and first-year graduate students of mathematics, statistics, probability, or engineering is revised for a new generation of students and instructors. The book also serves as a handy reference for professional mathematicians.

Real analysis

Author: G. B. Folland
Publisher: Wiley-Interscience
ISBN: UOM:49015000929258
Release Date: 1984-09
Genre: Mathematics

This book covers the subject matter that is central to mathematical analysis: measure & integration theory, some point set topology, & rudiments of functional analysis. Also, a number of other topics are developed to illustrate the uses of this core material in important areas of mathematics & to introduce readers to more advanced techniques. Some of the material presented has never appeared outside of advanced monographs & research papers, or been readily available in comparative texts. About 460 exercises, at varying levels of difficulty, give readers practice in working with the ideas presented here.

Techniques of Functional Analysis for Differential and Integral Equations

Author: Paul Sacks
Publisher: Academic Press
ISBN: 9780128114575
Release Date: 2017-05-16
Genre: Mathematics

Techniques of Functional Analysis for Differential and Integral Equations describes a variety of powerful and modern tools from mathematical analysis, for graduate study and further research in ordinary differential equations, integral equations, and especially partial differential equations. Knowledge of these techniques is particularly useful as preparation for graduate courses and as PhD research preparation in differential equations and numerical analysis, and more specialized topics such as fluid dynamics and control theory. Striking a balance between mathematical depth and accessibility, proofs are limited, and their sources precisely identifie d, proofs involving more technical aspects of measure and integration theory are avoided, but clear statements and precise alternative references are given . The work provides many examples and exercises drawn from the literature. Provides an introduction to the mathematical techniques widely used in applied mathematics and needed for advanced research Establishes the advanced background needed for sophisticated literature review and research in both differential and integral equations Suitable for use as a textbook for a two semester graduate level course for M.S. and Ph.D. students in Mathematics and Applied Mathematics

Advanced Real Analysis

Author: Anthony W. Knapp
Publisher: Springer Science & Business Media
ISBN: 0817644423
Release Date: 2008-07-11
Genre: Mathematics

* 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

Measure and Integration

Author: Leonard F. Richardson
Publisher: John Wiley & Sons
ISBN: 0470501146
Release Date: 2009-07-01
Genre: Mathematics

A uniquely accessible book for general measure and integration, emphasizing the real line, Euclidean space, and the underlying role of translation in real analysis Measure and Integration: A Concise Introduction to Real Analysis presents the basic concepts and methods that are important for successfully reading and understanding proofs. Blending coverage of both fundamental and specialized topics, this book serves as a practical and thorough introduction to measure and integration, while also facilitating a basic understanding of real analysis. The author develops the theory of measure and integration on abstract measure spaces with an emphasis of the real line and Euclidean space. Additional topical coverage includes: Measure spaces, outer measures, and extension theorems Lebesgue measure on the line and in Euclidean space Measurable functions, Egoroff's theorem, and Lusin's theorem Convergence theorems for integrals Product measures and Fubini's theorem Differentiation theorems for functions of real variables Decomposition theorems for signed measures Absolute continuity and the Radon-Nikodym theorem Lp spaces, continuous-function spaces, and duality theorems Translation-invariant subspaces of L2 and applications The book's presentation lays the foundation for further study of functional analysis, harmonic analysis, and probability, and its treatment of real analysis highlights the fundamental role of translations. Each theorem is accompanied by opportunities to employ the concept, as numerous exercises explore applications including convolutions, Fourier transforms, and differentiation across the integral sign. Providing an efficient and readable treatment of this classical subject, Measure and Integration: A Concise Introduction to Real Analysis is a useful book for courses in real analysis at the graduate level. It is also a valuable reference for practitioners in the mathematical sciences.

Reelle und Komplexe Analysis

Author: Walter Rudin
Publisher: Walter de Gruyter
ISBN: 348659186X
Release Date: 2009
Genre: Analysis - Lehrbuch

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.

Lebesgue Measure and Integration

Author: Frank Burk
Publisher: John Wiley & Sons
ISBN: 9781118030981
Release Date: 2011-10-14
Genre: Mathematics

A superb text on the fundamentals of Lebesgue measure and integration. This book is designed to give the reader a solid understanding of Lebesgue measure and integration. It focuses on only the most fundamental concepts, namely Lebesgue measure for R and Lebesgue integration for extended real-valued functions on R. Starting with a thorough presentation of the preliminary concepts of undergraduate analysis, this book covers all the important topics, including measure theory, measurable functions, and integration. It offers an abundance of support materials, including helpful illustrations, examples, and problems. To further enhance the learning experience, the author provides a historical context that traces the struggle to define "area" and "area under a curve" that led eventually to Lebesgue measure and integration. Lebesgue Measure and Integration is the ideal text for an advanced undergraduate analysis course or for a first-year graduate course in mathematics, statistics, probability, and other applied areas. It will also serve well as a supplement to courses in advanced measure theory and integration and as an invaluable reference long after course work has been completed.

An Introduction to Nonstandard Real Analysis

Author: Albert E. Hurd
Publisher: Academic Press
ISBN: 0080874371
Release Date: 1985-10-01
Genre: Mathematics

The aim of this book is to make Robinson's discovery, and some of the subsequent research, available to students with a background in undergraduate mathematics. In its various forms, the manuscript was used by the second author in several graduate courses at the University of Illinois at Urbana-Champaign. The first chapter and parts of the rest of the book can be used in an advanced undergraduate course. Research mathematicians who want a quick introduction to nonstandard analysis will also find it useful. The main addition of this book to the contributions of previous textbooks on nonstandard analysis (12,37,42,46) is the first chapter, which eases the reader into the subject with an elementary model suitable for the calculus, and the fourth chapter on measure theory in nonstandard models.

Real Analysis with an Introduction to Wavelets and Applications

Author: Don Hong
Publisher: Elsevier
ISBN: 0080540317
Release Date: 2004-12-31
Genre: Mathematics

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

A first course in functional analysis

Author: S. David Promislow
ISBN: 0470146192
Release Date: 2008-04-11
Genre: Mathematics

A concise introduction to the major concepts of functional analysis Requiring only a preliminary knowledge of elementary linear algebra and real analysis, A First Course in Functional Analysis provides an introduction to the basic principles and practical applications of functional analysis. Key concepts are illustrated in a straightforward manner, which facilitates a complete and fundamental understanding of the topic. This book is based on the author's own class-tested material and uses clear language to explain the major concepts of functional analysis, including Banach spaces, Hilbert spaces, topological vector spaces, as well as bounded linear functionals and operators. As opposed to simply presenting the proofs, the author outlines the logic behind the steps, demonstrates the development of arguments, and discusses how the concepts are connected to one another. Each chapter concludes with exercises ranging in difficulty, giving readers the opportunity to reinforce their comprehension of the discussed methods. An appendix provides a thorough introduction to measure and integration theory, and additional appendices address the background material on topics such as Zorn's lemma, the Stone-Weierstrass theorem, Tychonoff's theorem on product spaces, and the upper and lower limit points of sequences. References to various applications of functional analysis are also included throughout the book. A First Course in Functional Analysis is an ideal text for upper-undergraduate and graduate-level courses in pure and applied mathematics, statistics, and engineering. It also serves as a valuable reference for practitioners across various disciplines, including the physical sciences, economics, and finance, who would like to expand their knowledge of functional analysis.

Strange Functions in Real Analysis Second Edition

Author: A.B. Kharazishvili
Publisher: CRC Press
ISBN: 0824703200
Release Date: 2000-01-31
Genre: Mathematics

This volume aims to explicate extraordinary functions in real analysis and their applications. It examines the Baire category method, the Zermelo-Fraenkel set, the Axiom of Dependent Choices, Cantor and Peano type functions, the Continuum Hypothesis, everywhere differentiable nowhere monotone functions, and Jarnik's nowhere approximately differentiable functions.

Mathematical Analysis

Author: Bernd S. W. Schröder
Publisher: John Wiley & Sons
ISBN: 0470226765
Release Date: 2008-01-28
Genre: Mathematics

A self-contained introduction to the fundamentals of mathematical analysis Mathematical Analysis: A Concise Introduction presents the foundations of analysis and illustrates its role in mathematics. By focusing on the essentials, reinforcing learning through exercises, and featuring a unique "learn by doing" approach, the book develops the reader's proof writing skills and establishes fundamental comprehension of analysis that is essential for further exploration of pure and applied mathematics. This book is directly applicable to areas such as differential equations, probability theory, numerical analysis, differential geometry, and functional analysis. Mathematical Analysis is composed of three parts: ?Part One presents the analysis of functions of one variable, including sequences, continuity, differentiation, Riemann integration, series, and the Lebesgue integral. A detailed explanation of proof writing is provided with specific attention devoted to standard proof techniques. To facilitate an efficient transition to more abstract settings, the results for single variable functions are proved using methods that translate to metric spaces. ?Part Two explores the more abstract counterparts of the concepts outlined earlier in the text. The reader is introduced to the fundamental spaces of analysis, including Lp spaces, and the book successfully details how appropriate definitions of integration, continuity, and differentiation lead to a powerful and widely applicable foundation for further study of applied mathematics. The interrelation between measure theory, topology, and differentiation is then examined in the proof of the Multidimensional Substitution Formula. Further areas of coverage in this section include manifolds, Stokes' Theorem, Hilbert spaces, the convergence of Fourier series, and Riesz' Representation Theorem. ?Part Three provides an overview of the motivations for analysis as well as its applications in various subjects. A special focus on ordinary and partial differential equations presents some theoretical and practical challenges that exist in these areas. Topical coverage includes Navier-Stokes equations and the finite element method. Mathematical Analysis: A Concise Introduction includes an extensive index and over 900 exercises ranging in level of difficulty, from conceptual questions and adaptations of proofs to proofs with and without hints. These opportunities for reinforcement, along with the overall concise and well-organized treatment of analysis, make this book essential for readers in upper-undergraduate or beginning graduate mathematics courses who would like to build a solid foundation in analysis for further work in all analysis-based branches of mathematics.

Measure and Integration Theory

Author: Heinz Bauer
Publisher: Walter de Gruyter
ISBN: 3110167190
Release Date: 2001-01
Genre: Mathematics

This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the clear and approved exposition given in the author's earlier book on "Probability Theory and Measure Theory." Special emphasis is laid on a complete discussion of the transformation of measures and integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon-Nikodym theorem. The final chapter, essentially new and written in a clear and concise style, deals with the theory of Radon measures on Polish or locally compact spaces. With the main results being Luzin's theorem, the Riesz representation theorem, the Portmanteau theorem, and a characterization of locally compact spaces which are Polish, this chapter is a true invitation to study topological measure theory. The text addresses graduate students, who wish to learn the fundamentals in measure and integration theory as needed in modern analysis and probability theory. It will also be an important source for anyone teaching such a course.

Basic Real Analysis

Author: Anthony W. Knapp
Publisher: Springer Science & Business Media
ISBN: 9780817644413
Release Date: 2007-10-04
Genre: Mathematics

Systematically develop the concepts and tools that are vital to every mathematician, whether pure or applied, aspiring or established A comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics Included throughout are many examples and hundreds of problems, and a separate 55-page section gives hints or complete solutions for most.