*Mathematics | Program Guide: Real Analysis

This widely used analysis book covers real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory.

This volume is designed specifically for a one-semester course in real analysis. Many students of mathematics, physical and computer sciences need a text that presents the most important material in a brief and elementary fashion. The author meets this need with such elementary topics as the real number system, the theory at the basis of elementary calculus, the topology of metric spaces and infinite series. There are proofs of the basic theorems on limits at a pace that is deliberate and detailed, backed by illustrative examples throughout and no less than 45 figures.

Calculus answers questions that had been explored for centuries before calculus was born. Calculus and Its Origins begins with these ancient questions and details the remarkable story of how subsequent scholars wove these inquiries into a unified theory. This book does not presuppose knowledge of calculus, it requires only a basic knowledge of geometry and algebra (similar triangles, polynomials, factoring). Inside you will find the accounts of how Archimedes discovered the area of a parabolic segment, ibn Al-Haytham calculated the volume of a revolved area, Jyesthadeva explained the infinite series for sine and cosine, Wallis deduced the link between hyperbolas and logarithms, Newton generalized the binomial theorem, Leibniz discovered integration by parts, and much more. Each chapter ends with further results, in the form of exercises, by such luminaries as Pascal, Maclaurin, Barrow, Cauchy and Euler.

Counterexamples in Calculus serves as a supplementary resource to enhance the learning experience in single variable calculus courses. This book features carefully constructed incorrect mathematical statements that require students to create counterexamples to disprove them. Incorrect statements are grouped topically with sections devoted to: functions, limits, continuity, differential calculus and integral calculus. This book aims to fill a gap in the literature and provide a resource for using counterexamples as a pedagogical tool in the study of introductory calculus.

Calculus is the basis of all advanced science and math. But it can be very intimidating, especially if you're learning it for the first time! If finding derivatives or understanding integrals has you stumped, this book can guide you through it. This indispensable resource offers hundreds of practice exercises and covers all the key concepts of calculus. By breaking down challenging concepts and presenting clear explanations, you'll solidify your knowledge base--and face calculus without fear!

A complete course on metric, normed, and Hilbert spaces, including many results and exercises seldom found in texts on analysis at this level. The author covers an unusually wide range of material in a clear and concise format, including elementary real analysis, Lebesgue integration on R, and an introduction to functional analysis. This provides a reference for later chapters as well as a preparation for students with only the typical sequence of undergraduate calculus courses as prerequisites. Of special interest are the 750 exercises, many with guidelines for their solutions, applications and extensions of the main propositions and theorems, pointers to new branches of the subject, and difficult challenges for the very best students.

A uniquely accessible book for general measure and integration,emphasizing the real line, Euclidean space, and the underlying roleof translation in real analysis Measure and Integration: A Concise Introduction to RealAnalysis presents the basic concepts and methods that areimportant for successfully reading and understanding proofs.Blending coverage of both fundamental and specialized topics, thisbook serves as a practical and thorough introduction to measure andintegration, while also facilitating a basic understanding of realanalysis. The author develops the theory of measure and integration onabstract measure spaces with an emphasis of the real line andEuclidean 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'stheorem 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 dualitytheorems Translation-invariant subspaces of L2 and applications The book's presentation lays the foundation for further study offunctional analysis, harmonic analysis, and probability, and itstreatment of real analysis highlights the fundamental role oftranslations. Each theorem is accompanied by opportunities toemploy the concept, as numerous exercises explore applicationsincluding convolutions, Fourier transforms, and differentiationacross the integral sign. Providing an efficient and readable treatment of this classicalsubject, Measure and Integration: A Concise Introduction to RealAnalysis is a useful book for courses in real analysis at thegraduate level. It is also a valuable reference for practitionersin the mathematical sciences.

Presents a treatment of single variable Calculus designed as an introductory tertiary level mathematics textbook for engineering and science students. The subject matter is developed by modeling physical problems, some of which would normally be encountered by students as experiments in a first year physics course.

This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The new edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.

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, exercises and real world applications.