Lectures on Functional Analysis and the Lebesgue Integral / by Vilmos Komornik.

Format:

Book

Author/Creator:

Komornik, V., Author.

Series:

Universitext, 2191-6675

Language:

English

Subjects (All):

Functional analysis.

Measure theory.

Approximation theory.

Functional Analysis.

Measure and Integration.

Approximations and Expansions.

Local Subjects:

Functional Analysis.

Measure and Integration.

Approximations and Expansions.

Physical Description:

1 online resource (XX, 403 p. 46 illus.)

Edition:

1st ed. 2016.

Place of Publication:

London : Springer London : Imprint: Springer, 2016.

Summary:

This textbook, based on three series of lectures held by the author at the University of Strasbourg, presents functional analysis in a non-traditional way by generalizing elementary theorems of plane geometry to spaces of arbitrary dimension. This approach leads naturally to the basic notions and theorems. Most results are illustrated by the small ℓp spaces. The Lebesgue integral, meanwhile, is treated via the direct approach of Frigyes Riesz, whose constructive definition of measurable functions leads to optimal, clear-cut versions of the classical theorems of Fubini-Tonelli and Radon-Nikodým. Lectures on Functional Analysis and the Lebesgue Integral presents the most important topics for students, with short, elegant proofs. The exposition style follows the Hungarian mathematical tradition of Paul Erdős and others. The order of the first two parts, functional analysis and the Lebesgue integral, may be reversed. In the third and final part they are combined to study various spaces of continuous and integrable functions. Several beautiful, but almost forgotten, classical theorems are also included. Both undergraduate and graduate students in pure and applied mathematics, physics and engineering will find this textbook useful. Only basic topological notions and results are used and various simple but pertinent examples and exercises illustrate the usefulness and optimality of most theorems. Many of these examples are new or difficult to localize in the literature, and the original sources of most notions and results are indicated to help the reader understand the genesis and development of the field.

Contents:

Some papers of general interest

Topological prerequisites

Part 1 Functional analysis

Hilbert spaces

Banach spaces

Locally convex spaces

Part 2 The Lebesgue integral

Monotone functions.- The Lebesgue integral in R

Generalized Newton-Leibniz formula

Integrals on measure spaces

Part 3 Function spaces.- Spaces of continuous functions

Spaces of integrable functions

Almost everywhere convergence

Hints and solutions to some exercises.- Bibliography

Teaching remarks

Subject index

Name index.

Notes:

Includes bibliographical references and index.

ISBN:

1-4471-6811-9