Examining the basic principles in real analysis and their applications, this text provides a self-contained resource for graduate and advanced undergraduate courses. It contains independent chapters aimed at various fields of application, enhanced by highly advanced graphics and results explained and supplemented with practical and theoretical exercises. The presentation of the book is meant to provide natural connections to classical fields of applications such as Fourier analysis or statistics. However, the book also covers modern areas of research, including new and seminal results in the area of functional analysis.

This text is based on lectures given by the author at the advanced undergraduate and graduate levels in Measure Theory, Functional Analysis, Banach Algebras, Spectral Theory (of bounded and unbounded operators), Semigroups of Operators, Probability and Mathematical Statistics, and Partial Differential Equations. The first 10 chapters discuss theoretical methods in Measure Theory and Functional Analysis, and contain over 120 end of chapter exercises. The final two chapters apply theory to applications in Probability Theory and Partial Differential Equations. The Measure Theory chapters discuss the Lebesgue-Radon-Nikodym theorem which is given the Von Neumann Hilbert space proof. Also included are the relatively advanced topics of Haar measure, differentiability of complex Borel measures in Euclidean space with respect to Lebesgue measure, and the Marcinkiewicz' interpolation theorem for operators between Lebesgue spaces. The Functional Analysis chapters cover the usual material on Banach spaces, weak topologies, separation, extremal points, the Stone-Weierstrass theorem, Hilbert spaces, Banach algebras, and Spectral Theory for both bounded and unbounded operators. Relatively advanced topics such as the Gelfand-Naimark-Segal representation theorem and the Von Neumann double commutant theorem are included. The final two chapters deal with applications, where the measure theory and functional analysis methods of the first ten chapters are applied to Probability Theory and the Theory of Distributions and PDE's. Again, some advanced topics are included, such as the Lyapounov Central Limit theorem, the Kolmogoroff "Three Series theorem", the Ehrenpreis-Malgrange-Hormander theorem on fundamental solutions, and Hormander's theory of convolution operators. The Oxford Graduate Texts in Mathematics series aim is to publish textbooks suitable for graduate students in mathematics and its applications. The level of books may range from some suitable for advanced undergraduate courses at one end, to others of interest to research workers. The emphasis is on texts of high mathematical quality in active areas, particularly areas that are not well represented in the literature at present.

Aimed primarily at undergraduate level university students, An Illustrative Introduction to Modern Analysis provides an accessible and lucid contemporary account of the fundamental principles of Mathematical Analysis. The themes treated include Metric Spaces, General Topology, Continuity, Completeness, Compactness, Measure Theory, Integration, Lebesgue Spaces, Hilbert Spaces, Banach Spaces, Linear Operators, Weak and Weak* Topologies. Suitable both for classroom use and independent reading, this book is ideal preparation for further study in research areas where a broad mathematical toolbox is required.

Many applied mathematical disciplines, such as dynamical systems and optimization theory as well as classical mathematical disciplines like differential geometry and the theory of Lie groups, have a common foundation in general topology and multivariate calculus in normed vector spaces. In this book, students from both pure and applied subjects are offered an opportunity to work seriously with fundamental notions from mathematical analysis that are important not only from a mathematical point of view but also occur frequently in the theoretical parts of, for example, the engineering sciences. The book provides complete proofs of the basic results from topology and differentiability of mappings in normed vector spaces. It is a useful resource for students and researchers in mathematics and the many sciences that depend on fundamental techniques from mathematical analysis.In this second edition, the notions of compactness and sequentially compactness are developed with independent proofs for the main results. Thereby the material on compactness is apt for direct applications also in functional analysis, where the notion of sequentially compactness prevails. This edition also covers a new section on partial derivatives, and new material has been incorporated to make a more complete account of higher order derivatives in Banach spaces, including full proofs for symmetry of higher order derivatives and Taylor's formula. The exercise material has been reorganized from a collection of problem sets at the end of the book to a section at the end of each chapter with further results. Readers will find numerous new exercises at different levels of difficulty for practice.

A Passage to Modern Analysis is an extremely well-written and reader-friendly invitation to real analysis. An introductory text for students of mathematics and its applications at the advanced undergraduate and beginning graduate level, it strikes an especially good balance between depth of coverage and accessible exposition. The examples, problems, and exposition open up a student's intuition but still provide coverage of deep areas of real analysis. A yearlong course from this text provides a solid foundation for further study or application of real analysis at the graduate level. A Passage to Modern Analysis is grounded solidly in the analysis of R and Rn, but at appropriate points it introduces and discusses the more general settings of inner product spaces, normed spaces, and metric spaces. The last five chapters offer a bridge to fundamental topics in advanced areas such as ordinary differential equations, Fourier series and partial differential equations, Lebesgue measure and the Lebesgue integral, and Hilbert space. Thus, the book introduces interesting and useful developments beyond Euclidean space where the concepts of analysis play important roles, and it prepares readers for further study of those developments.

This classic text is known to and used by thousands of mathematicians and students of mathematics thorughout the world. It gives an introduction to the general theory of infinite processes and of analytic functions together with an account of the principle transcendental functions.

This book discusses some of the first principles of modern analysis. I t can be used for courses at several levels, depending upon the background and ability of the students. It was written on the premise that today's good students have unexpected enthusiasm and nerve. When hard work is put to them, they work harder and ask for more. The honors course (at the University of Wisconsin) which inspired this book was, I think, more fun than the book itself. And better. But then there is acting in teaching, and a typewriter is a poor substitute for an audience. The spontaneous, creative disorder that characterizes an exciting course becomes silly in a book. To write, one must cut and dry. Yet, I hope enough of the spontaneity, enough of the spirit of that course, is left to enable those using the book to create exciting courses of their own. Exercises in this book are not designed for drill. They are designed to clarify the meanings of the theorems, to force an understanding of the proofs, and to call attention to points in a proof that might otherwise be overlooked. The exercises, therefore, are a real part of the theory, not a collection of side issues, and as such nearly all of them are to be done. Some drill is, of course, necessary, particularly in the calculation of integrals.

Measure and integration, metric spaces, the elements of functional analysis in Banach spaces, and spectral theory in Hilbert spaces — all in a single study. Only book of its kind. Unusual topics, detailed analyses. Problems. Excellent for first-year graduate students, almost any course on modern analysis. Preface. Bibliography. Index.

Translated from the Russian revised and updated 1988 edition. Cubature formulas, for calculating the volumes of bodies in multidimensional space, were named by analogy with quadrature formulas, used to calculate the areas of plane figures. Topics include basic concepts and formulations, the polyharmonic equation, simple problems of the theory of computations, order of convergence of cubature formulas, considering a regular boundary layer, optimal formulas, and formulas for rational polyhedra. Annotation copyright by Book News, Inc., Portland, OR

Modern Analysis provides coverage of real and abstract analysis, offering a sensible introduction to functional analysis as well as a thorough discussion of measure theory, Lebesgue integration, and related topics. This significant study clearly and distinctively presents the teaching and research literature of graduate analysis: Providing a fundamental, modern approach to measure theory Investigating advanced material on the Bochner integral, geometric theory, and major theorems in Fourier Analysis Rn, including the theory of singular integrals and Milhin's theorem - material that does not appear in textbooks Offering exceptionally concise and cardinal versions of all the main theorems about characteristic functions Containing an original examination of sufficient statistics, based on the general theory of Radon measures With an ambitious scope, this resource unifies various topics into one volume succinctly and completely. The contents span basic measure theory in an abstract and concrete form, material on classic linear functional analysis, probability, and some major results used in the theory of partial differential equations. Two different proofs of the central limit theorem are examined as well as a straightforward approach to conditional probability and expectation. Modern Analysis provides ample and well-constructed exercises and examples. Introductory topology is included to help the reader understand such items as the Riesz theorem, detailing its proofs and statements. This work will help readers apply measure theory to probability theory, guiding them to understand the theorems rather than merely follow directions.