Convergence Problems For Functionals And Operators
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Functional Analysis and Operator Theory
The book contains a collection of more than 800 problems from all main chapters of functional analysis, with theoretical background and solutions. It is mostly intended for undergraduate students who are starting to study the course of functional analysis. The book will also be useful for graduate and post- graduate students and researchers who wish to refresh their knowledge and deepen their understanding of the subject, as well as for teachers of functional analysis and related disciplines. It can be used for independent study as well. It is assumed that the reader has mastered standard courses of calculus and measure theory and has basic knowledge of linear algebra, analytic geometry, and differential equations. This collection of problems can help students of different levels of training and different areas of specialization to learn how to solve problems in functional analysis. Each chapter of the book has similar structure and consists of the following sections: Theoretical Background, Examples of Problems with Solutions, and Problems to Solve. The book contains theoretical preliminaries to ensure that the reader understands the statements of problems and is able to successfully solve them. Then examples of typical problems with detailed solutions are included, and this is relevant not only for those students who have significant difficulties in studying this subject, but also for other students who due to various circumstances сcould be deprived of communication with a teacher. There are problems for independent solving, and the corresponding selection of problems reflects all the main plot lines that relate to a given topic. The number of problems is sufficient both for a teacher to give practical lessons, to set homework, to prepare tasks for various forms of control, and for those students who want to study the discipline more deeply. Problems of a computational nature are provided with answers, while theoretical problems, the solutions ofwhich require non-trivial ideas or new techniques, are provided with detailed hints or solutions to introduce the reader to the corresponding ideas or techniques.
Theorems and Problems in Functional Analysis
Author: A. A. Kirillov
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-12-06
Even the simplest mathematical abstraction of the phenomena of reality the real line-can be regarded from different points of view by different mathematical disciplines. For example, the algebraic approach to the study of the real line involves describing its properties as a set to whose elements we can apply" operations," and obtaining an algebraic model of it on the basis of these properties, without regard for the topological properties. On the other hand, we can focus on the topology of the real line and construct a formal model of it by singling out its" continuity" as a basis for the model. Analysis regards the line, and the functions on it, in the unity of the whole system of their algebraic and topological properties, with the fundamental deductions about them obtained by using the interplay between the algebraic and topological structures. The same picture is observed at higher stages of abstraction. Algebra studies linear spaces, groups, rings, modules, and so on. Topology studies structures of a different kind on arbitrary sets, structures that give mathe matical meaning to the concepts of a limit, continuity, a neighborhood, and so on. Functional analysis takes up topological linear spaces, topological groups, normed rings, modules of representations of topological groups in topological linear spaces, and so on. Thus, the basic object of study in functional analysis consists of objects equipped with compatible algebraic and topological structures.
Lie Algebras, Vertex Operator Algebras, and Related Topics
Author: Katrina Barron
language: en
Publisher: American Mathematical Soc.
Release Date: 2017-08-15
This volume contains the proceedings of the conference on Lie Algebras, Vertex Operator Algebras, and Related Topics, celebrating the 70th birthday of James Lepowsky and Robert Wilson, held from August 14–18, 2015, at the University of Notre Dame, Notre Dame, Indiana. Since their seminal work in the 1970s, Lepowsky and Wilson, their collaborators, their students, and those inspired by their work, have developed an amazing body of work intertwining the fields of Lie algebras, vertex algebras, number theory, theoretical physics, quantum groups, the representation theory of finite simple groups, and more. The papers presented here include recent results and descriptions of ongoing research initiatives representing the broad influence and deep connections brought about by the work of Lepowsky and Wilson and include a contribution by Yi-Zhi Huang summarizing some major open problems in these areas, in particular as they pertain to two-dimensional conformal field theory.