This is the second edition of a graduate level real analysis textbook formerly published by Prentice Hall (Pearson) in 1997. This edition contains both volumes. Volumes one and two can also be purchased separately in smaller, more convenient sizes.
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.
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.
This revised edition provides an excellent introduction to topics in Real Analysis through an elaborate exposition of all fundamental concepts and results. The treatment is rigorous and exhaustive—both classical and modern topics are presented in a lucid manner in order to make this text appealing to students. Clear explanations, many detailed worked examples and several challenging ones included in the exercises, enable students to develop problem-solving skills and foster critical thinking. The coverage of the book is incredibly comprehensive, with due emphasis on Lebesgue theory, metric spaces, uniform convergence, Riemann–Stieltjes integral, multi-variable theory, Fourier series, improper integration, and parametric integration. The book is suitable for a complete course in real analysis at the advanced undergraduate or postgraduate level.
Real Analysis and Foundations is an advanced undergraduate and first-year graduate textbook that introduces students to introductory topics in real analysis (or real variables), point set topology, and the calculus of variations. This classroom-tested book features over 350 end-of-chapter exercises that clearly develop and reinforce conceptual topics. It also provides an excellent review chapter on math foundations topics, as well as accessible coverage of classical topics, such as Weirstrass Approximation Theorem, Ascoli-Arzela Theorem and Schroeder-Bernstein Theorem. Explanations and discussions of key concepts are so well done that Real Analysis and Foundations will also provide valuable information for professional aerospace and structural engineers.
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.
This classic graduate textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The comprehensive historical notes have been further amplified for this new edition, and a number of new exercises have been added, together with hints for solution.
An in-depth look at real analysis and its applications-now expandedand revised. This new edition of the widely used analysis book continues tocover real analysis in greater detail and at a more advanced levelthan most books on the subject. Encompassing several subjects thatunderlie much of modern analysis, the book focuses on measure andintegration theory, point set topology, and the basics offunctional analysis. It illustrates the use of the general theoriesand introduces readers to other branches of analysis such asFourier analysis, distribution theory, and probabilitytheory. This edition is bolstered in content as well as in scope-extendingits usefulness to students outside of pure analysis as well asthose interested in dynamical systems. The numerous exercises,extensive bibliography, and review chapter on sets and metricspaces make Real Analysis: Modern Techniques and TheirApplications, Second Edition invaluable for students ingraduate-level analysis courses. New features include: * Revised material on the n-dimensional Lebesgue integral. * An improved proof of Tychonoff's theorem. * Expanded material on Fourier analysis. * A newly written chapter devoted to distributions and differentialequations. * Updated material on Hausdorff dimension and fractal dimension.