Functional Analysis: Introduction to Further Topics in Analysis (Princeton Lectures in Analysis, 4, Band 4) - Hardcover

Stein, Elias M.; Shakarchi, Rami

Hardcover

ISBN 10: 0691113874 ISBN 13: 9780691113876

Verlag: Princeton University Press, 2011

Inhaltsangabe

This is the fourth and final volume in the Princeton Lectures in Analysis, a series of textbooks that aim to present, in an integrated manner, the core areas of analysis. Beginning with the basic facts of functional analysis, this volume looks at Banach spaces, Lp spaces, and distribution theory, and highlights their roles in harmonic analysis. The authors then use the Baire category theorem to illustrate several points, including the existence of Besicovitch sets. The second half of the book introduces readers to other central topics in analysis, such as probability theory and Brownian motion, which culminates in the solution of Dirichlet's problem. The concluding chapters explore several complex variables and oscillatory integrals in Fourier analysis, and illustrate applications to such diverse areas as nonlinear dispersion equations and the problem of counting lattice points. Throughout the book, the authors focus on key results in each area and stress the organic unity of the subject.

• A comprehensive and authoritative text that treats some of the main topics of modern analysis


• A look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables


• Key results in each area discussed in relation to other areas of mathematics


• Highlights the organic unity of large areas of analysis traditionally split into subfields


• Interesting exercises and problems illustrate ideas


• Clear proofs provided

Über diesen Titel

Über die Autorin bzw. den Autor

Elias M. Stein is the Albert Baldwin Dod Professor of Mathematics at Princeton University. Rami Shakarchi received his PhD in mathematics from Princeton University. They are the coauthors of Complex Analysis, Fourier Analysis, and Real Analysis (all Princeton).

Von der hinteren Coverseite

"This book introduces basic functional analysis, probability theory, and most importantly, aspects of modern analysis that have developed over the last half century. It is the first student-oriented textbook where all of these topics are brought together with lots of interesting exercises and problems. This is a valuable addition to the literature."--Gerald B. Folland, University of Washington

Aus dem Klappentext

"This book introduces basic functional analysis, probability theory, and most importantly, aspects of modern analysis that have developed over the last half century. It is the first student-oriented textbook where all of these topics are brought together with lots of interesting exercises and problems. This is a valuable addition to the literature."--Gerald B. Folland, University of Washington

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FUNCTIONAL ANALYSIS

Introduction to Further Topics in AnalysisBy Elias M. Stein Rami Shakarchi

PRINCETON UNIVERSITY PRESS

Copyright © 2011 Princeton University Press
All right reserved.
ISBN: 978-0-691-11387-6

Contents

Foreword..................................................................viiPreface...................................................................xviiChapter 1. Lp Spaces and Banach Spaces.........................1Chapter 2. Lp Spaces in Harmonic Analysis......................47Chapter 3. Distributions: Generalized Functions...........................98Chapter 4. Applications of the Baire Category Theorem.....................157Chapter 5. Rudiments of Probability Theory................................188Chapter 6. An Introduction to Brownian Motion.............................238Chapter 7. A Glimpse into Several Complex Variables.......................276Chapter 8. Oscillatory Integrals in Fourier Analysis......................321Notes and References......................................................409Bibliography..............................................................413Symbol Glossary...........................................................417Index.....................................................................419

Chapter One

L^p Spaces and Banach Spaces

In this work the assumption of quadratic integrability will be replaced by the integrability of |f(x)|^p. The analysis of these function classes will shed a particular light on the real and apparent advantages of the exponent 2; one can also expect that it will provide essential material for an axiomatic study of function spaces.

F. Riesz, 1910

At present I propose above all to gather results about linear operators defined in certain general spaces, notably those that will here be called spaces of type (B) ...

S. Banach, 1932

Function spaces, in particular L^p spaces, play a central role in many questions in analysis. The special importance of L^p spaces may be said to derive from the fact that they offer a partial but useful generalization of the fundamental L² space of square integrable functions.

In order of logical simplicity, the space L¹ comes first since it occurs already in the description of functions integrable in the Lebesgue sense. Connected to it via duality is the L^∞ space of bounded functions, whose supremum norm carries over from the more familiar space of continuous functions. Of independent interest is the L² space, whose origins are tied up with basic issues in Fourier analysis. The intermediate L^p spaces are in this sense an artifice, although of a most inspired and fortuitous kind. That this is the case will be illustrated by results in the next and succeeding chapters.

In this chapter we will concentrate on the basic structural facts about the L^p spaces. Here part of the theory, in particular the study of their linear functionals, is best formulated in the more general context of Banach spaces. An incidental benefit of this more abstract view-point is that it leads us to the surprising discovery of a finitely additive measure on all subsets, consistent with Lebesgue measure.

1 L^p spaces

Throughout this chapter (X, F, μ) denotes a σ-finite measure space: X denotes the underlying space, F the σ-algebra of measurable sets, and μ the measure. If 1 ≤ p < ∞, the space L^p(X,F, μ) consists of all complex-valued measurable functions on X that satisfy

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

To simplify the notation, we write L^p(X, μ), or L^p(X), or simply L^p when the underlying measure space has been specified. Then, if f [member of] L^p(X,F, μ) we define the L^p norm of f by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We also abbreviate this to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

When p = 1 the space L¹(X,F, μ) consists of all integrable functions on X, and we have shown in Chapter 6 of Book III, that L¹ together with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a complete normed vector space. Also, the case p = 2 warrants special attention: it is a Hilbert space.

We note here that we encounter the same technical point that we already discussed in Book III. The problem is that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] does not imply that f = 0, but merely f = 0 almost everywhere (for the measure μ). Therefore, the precise definition of L^p requires introducing the equivalence relation, in which f and g are equivalent if f = g a.e. Then, L^p consists of all equivalence classes of functions which satisfy (1). However, in practice there is little risk of error by thinking of elements in L^p as functions rather than equivalence classes of functions.

The following are some common examples of L^p spaces.

(a) The case X = R^d and μ equals Lebesgue measure is often used in practice. There, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(b) Also, one can take X = Z, and μ equal to the counting measure. Then, we get the "discrete" version of the L^p spaces. Measurable functions are simply sequences f = {a_n}_{n[member of]Z} of complex numbers,

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

When p = 2, we recover the familiar sequence space l²(Z).

The spaces L^p are examples of normed vector spaces. The basic property satisfied by the norm is the triangle inequality, which we shall prove shortly.

The range of p which is of interest in most applications is 1 ≤ p < ∞, and later also p = ∞. There are at least two reasons why we restrict our attention to these values of p: when 0 < p < 1, the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] does not satisfy the triangle inequality, and moreover, for such p, the space L^p has no non-trivial bounded linear functionals. (See Exercise 2.)

When p = 1 the norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] satisfies the triangle inequality, and L¹ is a complete normed vector space. When p = 2, this result continues to hold, although one needs the Cauchy-Schwarz inequality to prove it. In the same way, for 1 ≤ p < ∞ the proof of the triangle inequality relies on a generalized version of the Cauchy-Schwarz inequality. This is Hölder's inequality, which is also the key in the duality of the L^p spaces, as we will see in Section 4.

1.1 The Hölder and Minkowski inequalities

If the two exponents p and q satisfy 1 ≤ p, q ≤ ∞, and the...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2011
• Sprache: Englisch
• ISBN 10: 0691113874
• ISBN 13: 9780691113876
• Einband: Gebundene Ausgabe
• Anzahl der Seiten: 423
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar