Shmuel Kantorovitz – författare

This textbook provides an introduction to modern analysis aimed at advanced undergraduate and graduate-level students of mathematics. Professional academics will also find this to be a useful reference work. It covers measure theory, basic functional analysis, single operator theory, spectral theory of bounded and unbounded operators, semigroups of operators, and Banach algebras. Further, this new edition of the textbook also delves deeper into C*-algebras and their standard constructions, von Neumann algebras, probability and mathematical statistics, and partial differential equations. Most chapters contain relatively advanced topics alongside simpler ones, starting from the very basics of modern analysis and slowly advancing to more involved topics. The text is supplemented by many exercises, to allow readers to test their understanding and practical analysis skills.

This textbook provides an introduction to modern analysis aimed at advanced undergraduate and graduate-level students of mathematics. Professional academics will also find this to be a useful reference work. It covers measure theory, basic functional analysis, single operator theory, spectral theory of bounded and unbounded operators, semigroups of operators, and Banach algebras. Further, this new edition of the textbook also delves deeper into C*-algebras and their standard constructions, von Neumann algebras, probability and mathematical statistics, and partial differential equations. Most chapters contain relatively advanced topics alongside simpler ones, starting from the very basics of modern analysis and slowly advancing to more involved topics. The text is supplemented by many exercises, to allow readers to test their understanding and practical analysis skills.

This textbook provides an introduction to modern analysis aimed at advanced undergraduate and graduate-level students of mathematics. Professional academics will also find this to be a useful reference work. It covers measure theory, basic functional analysis, single operator theory, spectral theory of bounded and unbounded operators, semigroups of operators, and Banach algebras. Further, this new edition of the textbook also delves deeper into C*-algebras and their standard constructions, von Neumann algebras, probability and mathematical statistics, and partial differential equations. Most chapters contain relatively advanced topics alongside simpler ones, starting from the very basics of modern analysis and slowly advancing to more involved topics. The text is supplemented by many exercises, to allow readers to test their understanding and practical analysis skills.

This book is based on lecture notes from a second-year graduate course, and is a greatly expanded version of our previous monograph [K8]. We expose some aspects of the theory of semigroups of linear operators, mostly (but not only) from the point of view of its meeting with that part of spectral theory which is concerned with the integral representation of families of operators. This approach and selection of topics di?erentiate this book from others in the general area, and re?ect the author’s own research directions. There is no attempt therefore to cover thoroughly the theory of semigroups of operators. This theory and its applications are extensively exposed in many books, from theclassicHille–Phillipsmonograph[HP]tothemostrecenttextbookofEngel and Nagel [EN2] (see [A], [BB], [Cl], [D3], [EN1], [EN2], [Fat], [G], [HP], [P], [Vr], and others), as well as in chapters in more general texts on Functional Analysis and the theory of linear operators (cf. [D5], [DS I–III], [Kat1], [RS], [Y], and many others).

This book is based on lecture notes from a second-year graduate course, and is a greatly expanded version of our previous monograph [K8]. We expose some aspects of the theory of semigroups of linear operators, mostly (but not only) from the point of view of its meeting with that part of spectral theory which is concerned with the integral representation of families of operators. This approach and selection of topics di?erentiate this book from others in the general area, and re?ect the author’s own research directions. There is no attempt therefore to cover thoroughly the theory of semigroups of operators. This theory and its applications are extensively exposed in many books, from theclassicHille–Phillipsmonograph[HP]tothemostrecenttextbookofEngel and Nagel [EN2] (see [A], [BB], [Cl], [D3], [EN1], [EN2], [Fat], [G], [HP], [P], [Vr], and others), as well as in chapters in more general texts on Functional Analysis and the theory of linear operators (cf. [D5], [DS I–III], [Kat1], [RS], [Y], and many others).

This undergraduate textbook is based on lectures given by the author on the differential and integral calculus of functions of several real variables. The book has a modern approach and includes topics such as:•The p-norms on vector space and their equivalence•The Weierstrass and Stone-Weierstrass approximation theorems•The differential as a linear functional; Jacobians, Hessians, and Taylor's theorem in several variables•The Implicit Function Theorem for a system of equations, proved via Banach’s Fixed Point Theorem•Applications to Ordinary Differential Equations•Line integrals and an introduction to surface integralsThis book features numerous examples, detailed proofs, as well as exercises at the end of sections. Many of the exercises have detailed solutions, making the book suitable for self-study. Several Real Variables will be useful for undergraduate students in mathematics who have completed first courses in linear algebra and analysis of one real variable.