Theorems and Problems in Functional Analysis

1. Front Matter

   Pages i-ix

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2. Theory

   1. Front Matter

      Pages 1-1

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   2. Concepts from Set Theory and Topology

      • A. A. Kirillov, A. A. Gvishiani

      Pages 3-11

   3. Theory of Measures and Integrals

      • A. A. Kirillov, A. A. Gvishiani

      Pages 12-37

   4. Linear Topological Spaces and Linear Operators

      • A. A. Kirillov, A. A. Gvishiani

      Pages 38-94

   5. The Fourier Transformation and Elements of Harmonic Analysis

      • A. A. Kirillov, A. A. Gvishiani

      Pages 95-115

   6. The Spectral Theory of Operators

      • A. A. Kirillov, A. A. Gvishiani

      Pages 116-135

3. Problems

   1. Front Matter

      Pages 137-137

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   2. Concepts from Set Theory and Topology

      • A. A. Kirillov, A. A. Gvishiani

      Pages 139-149

   3. Theory of Measures and Integrals

      • A. A. Kirillov, A. A. Gvishiani

      Pages 150-169

   4. Linear Topological Spaces and Linear Operators

      • A. A. Kirillov, A. A. Gvishiani

      Pages 170-203

   5. The Fourier Transformation and Elements of Harmonic Analysis

      • A. A. Kirillov, A. A. Gvishiani

      Pages 204-218

   6. The Spectral Theory of Operators

      • A. A. Kirillov, A. A. Gvishiani

      Pages 219-230

4. Hints

   1. Front Matter

      Pages 231-231

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   2. Concepts from Set Theory and Topology

      • A. A. Kirillov, A. A. Gvishiani

      Pages 233-243

   3. Theory of Measures and Integrals

      • A. A. Kirillov, A. A. Gvishiani

      Pages 244-270

   4. Linear Topological Spaces and Linear Operators

      • A. A. Kirillov, A. A. Gvishiani

      Pages 271-308

   5. The Fourier Transformation and Elements of Harmonic Analysis

      • A. A. Kirillov, A. A. Gvishiani

      Pages 309-324

   6. The Spectral Theory of Operators

      • A. A. Kirillov, A. A. Gvishiani

      Pages 325-334

5. Back Matter

   Pages 335-347

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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.

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

• Mathemathics Department, Moscow State University, Moscow, USSR

  A. A. Kirillov

• Applied Mathemathics Laboratory, Institute of Earth Physics of the Academy of Sciences of the USSR, Moscow, USSR

  A. A. Gvishiani

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