# Functional Analysis I

## Linear Functional Analysis

• Yu. I. Lyubich
• N. K. Nikol’skij
Book

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 19)

1. Front Matter
Pages I-5
2. Yu. I. Lyubich
Pages 6-84
3. Yu. I. Lyubich
Pages 85-260
4. Back Matter
Pages 261-286

### Introduction

Up to a certain time the attention of mathematicians was concentrated on the study of individual objects, for example, specific elementary functions or curves defined by special equations. With the creation of the method of Fourier series, which allowed mathematicians to work with 'arbitrary' functions, the individual approach was replaced by the 'class' approach, in which a particular function is considered only as an element of some 'function space'. More or less simultane­ ously the development of geometry and algebra led to the general concept of a linear space, while in analysis the basic forms of convergence for series of functions were identified: uniform, mean square, pointwise and so on. It turns out, moreover, that a specific type of convergence is associated with each linear function space, for example, uniform convergence in the case of the space of continuous functions on a closed interval. It was only comparatively recently that in this connection the general idea of a linear topological space (L TS)l was formed; here the algebraic structure is compatible with the topological structure in the sense that the basic operations (addition and multiplication by a scalar) are continuous.

### Keywords

Anwendungen der Funktionalalysis Applications of Functional Analysis Linear Functional Analysis Lineare Funktionalanalysis Operator theory Topological Vector Spaces Topologischer Vektorraum calculus differential equation functional analysis

#### Authors and affiliations

• Yu. I. Lyubich
• 1
1. 1.Department of MathematicsState University of New York at Stony BrookStony BrookUSA

### Editors and affiliations

• N. K. Nikol’skij
• 1
1. 1.Steklov Mathematical InstituteSt. PeterburgUSSR

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-662-02849-0
• Copyright Information Springer-Verlag Berlin Heidelberg 1992
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-08070-8
• Online ISBN 978-3-662-02849-0
• Series Print ISSN 0938-0396
• Buy this book on publisher's site