Spaces of Fundamental and Generalized Functions

Spaces of Fundamental and Generalized Functions, Volume 2, analyzes the general theory of linear topological spaces. The basis of the theory of generalized functions is the theory of the so-called countably normed spaces (with compatible norms), their unions (inductive limits), and also of the spaces conjugate to the countably normed ones or their unions. This set of spaces is sufficiently broad on the one hand, and sufficiently convenient for the analyst on the other.
The book opens with a chapter that discusses the theory of these spaces. This is followed by separate chapters on fundamental and generalized functions, Fourier transformations of fundamental and generalized functions, and spaces of type S.

Preface to the Russian Edition

Chapter I Linear Topological Spaces

1. Definition of a Linear Topological Space

2. Normed Spaces. Comparability and Compatibility of Norms

3. Countably Normed Spaces

4. Continuous Linear Functional and the Conjugate Space

5. Topology in a Conjugate Space

6. Perfect Spaces

7. Continuous Linear Operators

8. Union of Countably Normed Spaces

Appendix 1. Elements, Functionals, Operators Depending on a Parameter

Appendix 2. Differentiate Abstract Functions

Appendix 3. Operators Depending on a Parameter

Appendix 4. Integration of Continuous Abstract Functions with Respect to the Parameter

Chapter II Fundamental and Generalized Functions

1. Definition of Fundamental and Generalized Functions

2. Topology in the Spaces K{Mp} and Z{Mp}

3. Operations with Generalized Functions

4. Structure of Generalized Functions

Chapter III Fourier Transformations of Fundamental and Generalized Functions

1. Fourier Transformations of Fundamental Functions

2. Fourier Transforms of Generalized Functions

3. Convolution of Generalized Functions and Its Connection to Fourier Transforms

4. Fourier Transformation of Entire Analytic Functions

Chapter IV Spaces of Type S

1. Introduction

2. Various Modes of Defining Spaces of Type S

3. Topological Structure of Fundamental Spaces

4. Simplest Bounded Operations in Spaces of Type S

5. Differential Operators

6. Fourier Transformations

7. Entire Analytic Functions as Elements or Multipliers in Spaces of Type S

8. The Question of the Nontriviality of Spaces of Type S

9. The Case of Several Independent Variables

Appendix 1. Generalization of Spaces of Type S

Appendix 2. Spaces of Type W

Notes and References

Bibliography

Index