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Functional Analysis: Theory and Applicationsкнига
- Автор: Садовничий В.А.
- Год издания: 2019
- Место издания: ILS Publishing, Global Publication Company Hong Kong
- Объём: 427 страниц
- ISBN: 978-988-79343-3-2
- Учебник
- Аннотация: The textbook contains fundamental concepts and theorems of set theory, a general theory of metric, topological, topological vector and normed spaces, a general theory of measure, measurable functions and the Lebesgue integral, the theory of Fourier transforms, including generalized functions. For the first time in a textbook of this kind the material is introduced on the spectral theory of operators, the theory of traces of operators. The content of the textbook is as follows. Chapter 1 contains topical material on metric and topological spaces. Chapter 2 is devoted to vector spaces. Three basic principles of functional analysis are presented in the section on F-spaces. Now this chapter contains a detailed up-to-date material on convex sets in vector spaces and topological vector spaces. Chapter 3 is devoted to measure theory and the Lebesgue integral. The section of the Radon—Nikodym theorem and the Fubini theorem completes this chapter. Chapter 4 is devoted to the spectral theory of operators. We consider the properties of a Hilbert space, prove spectral theorems for a bounded self-adjoint operator, and also a theorem on the spectral decomposition of a self-adjoint unbounded operator in a Hilbert space. This edition of the textbook proves the statements that were introduced in the first edition as exercises. Chapter 5, The Trace of an Operator, contains proofs of theorems for trace of positive nuclear operator. In addition, a complete proof of Lidskii's theorem for a nuclear operator is given. Then the theory of regularized traces for the so-called functions of the class K is presented. The theorem for regularized traces for certain classes of discrete operators completes the chapter. These new results can be used to find regularized traces of partial differential operators. This material can be singled out into a separate course. Many propositions are illustrated by examples. Chapter 6 is devoted to the modern theory of Fourier transforms, including for generalized functions. The appropriate examples are given.
- Добавил в систему: Попеленский Михаил Юрьевич