# Theorems and Problems in Functional Analysis

• A. A. Kirillov
• A. A. Gvishiani
Textbook

Part of the Problem Books in Mathematics book series (PBM)

1. Front Matter
Pages i-ix
2. ### Theory

1. Front Matter
Pages 1-1
2. A. A. Kirillov, A. A. Gvishiani
Pages 3-11
3. A. A. Kirillov, A. A. Gvishiani
Pages 12-37
4. A. A. Kirillov, A. A. Gvishiani
Pages 38-94
5. A. A. Kirillov, A. A. Gvishiani
Pages 95-115
6. A. A. Kirillov, A. A. Gvishiani
Pages 116-135
3. ### Problems

1. Front Matter
Pages 137-137
2. A. A. Kirillov, A. A. Gvishiani
Pages 139-149
3. A. A. Kirillov, A. A. Gvishiani
Pages 150-169
4. A. A. Kirillov, A. A. Gvishiani
Pages 170-203
5. A. A. Kirillov, A. A. Gvishiani
Pages 204-218
6. A. A. Kirillov, A. A. Gvishiani
Pages 219-230
4. ### Hints

1. Front Matter
Pages 231-231
2. A. A. Kirillov, A. A. Gvishiani
Pages 233-243
3. A. A. Kirillov, A. A. Gvishiani
Pages 244-270
4. A. A. Kirillov, A. A. Gvishiani
Pages 271-308
5. A. A. Kirillov, A. A. Gvishiani
Pages 309-324
6. A. A. Kirillov, A. A. Gvishiani
Pages 325-334
5. Back Matter
Pages 335-347

### Introduction

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.

### Keywords

Convexity Fourier series Funktionalanalysis Funktionalanalysis /Aufgabensammlung Hilbert space Theorems calculus convolution differential equation functional analysis spectral theorem

#### Authors and affiliations

• A. A. Kirillov
• 1
• A. A. Gvishiani
• 2
1. 1.Mathemathics DepartmentMoscow State UniversityMoscowUSSR
2. 2.Applied Mathemathics LaboratoryInstitute of Earth Physics of the Academy of Sciences of the USSRMoscowUSSR

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4613-8153-2
• Copyright Information Springer-Verlag New York Inc. 1982
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4613-8155-6
• Online ISBN 978-1-4613-8153-2
• Series Print ISSN 0941-3502
• Series Online ISSN 2197-8506
• Buy this book on publisher's site