Elements of the Theory of Functions and Functional Analysis, Vol. I Metric and Normed Spaces

by Kolmogorov, A. N.; Fomin, S. V.

• ISBN13: 9780486697598
• ISBN10: 0486697592
• Format: Hardcover
• Copyright: 1996-09-13
• Publisher: Dover Publications

Summary

Concise, rigorous presentation of most of the elements of the theory of metric spaces and normed linear spaces. Discussions of set theory, measure and the Lebesgue integral, theory of functions of a real variable, linear operator equations, more. Translated from the first (1954) Russian edition. Bibliography.

Table of Contents

Preface

Translator's Note

Fundamentals of Set Theory

The concept of Set

Operations on Sets

Finite and Infinite Sets

Denumerability

Equivalence of Sets

The Nondenumerability of the Set of Real Numbers

The Concept of Cardinal Number

Partition into Classes

Mappings of Sets

General Concept of Function

Metric Spaces

Definition and Examples of Metric Spaces

Convergence of Sequences

Limit Points

Open and Closed Sets

Open and Closed Sets on the Real Line

Continuous Mappings

Homeomorphism

Isometry

Complete Metric Spaces

The Principle of Contraction Mappings and its Applications

Applications of the Principle of Contraction Mappings in Analysis

Compact Sets in Metric Spaces

Arzela's Theorem and its Applications

Compacta

Real functions in Metric Spaces

Continuous Curves in Metric Spaces

Normed Linear Spaces

Definition and Examples of Normed Linear Spaces

Convex Sets in Normed Linear Spaces

Linear Functionals

The Conjugate Space

Extension of Linear Functionals

The Second Conjugate Space

Weak Convergence

Weak Convergence of Linear Functionals

Linear Operators Addendum to

Generalized Functions

Linear Operator Equations

Spectrum of an Operator

Resolvents

Completely Continuous Operators

Linear Operator Equations

Fredholm's Theorems List of Symbols

List of Definitions

List of Theorems

Basic Literature

Index

Table of Contents provided by Publisher. All Rights Reserved.