Functional Analysis: Introduction to Further Topics in Analysis: 04 (Princeton Lectures in Analysis, 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).

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"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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"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 relation

1/p + 1/q = 1

holds, we say that p and q are conjugate or dual exponents. Here, we use the convention 1/∞ = 0. Later, we shall sometimes use p' to denote the conjugate exponent of p. Note that p = 2 is self-dual, that is, p = q = 2; also p = 1, ∞ corresponds to q = ∞, 1 respectively.

Theorem 1.1 (Hölder) Suppose 1 < p < ∞ and 1 < q < ∞ are conjugate exponents. If f [member of] L^p and g [member of] L^q, then fg [member of] L¹ and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note. Once we have defined L^∞ (see Section 2) the corresponding inequality for the exponents 1 and ∞ will be seen to be essentially trivial.

The proof of the theorem relies on a simple generalized form of the arithmetic-geometric mean inequality: if A, B ≥ 0, and 0 ≤ θ ≤ 1, then

(2) AθB1-θ ≤ θA + (1 - θ)B.

Note that when θ = 1/2, the inequality (2) states the familiar fact that the geometric mean of two numbers is majorized by their arithmetic mean.

To establish (2), we observe first that we may assume B [not equal to] 0, and replacing A by AB, we see that it suffices to prove that A^θ ≤ θA + (1 - θ). If we let f(x) = x^θ - θx - (1 - θ), then f'(x) = θ(x^θ-1 - 1). Thus f(x) increases when 0 ≤ x ≤ 1 and decreases when 1 ≤ x, and we see that the continuous function f attains a maximum at x = 1, where f(1) = 0. Therefore f(A) ≤ 0, as desired.

To prove Hölder's inequality we argue as follows. If either [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then fg = 0 a.e. and the inequality is obviously verified. Therefore, we may assume that neither of these norms vanish, and after replacing f by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and g by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we may further assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We now need to prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we set A = |f(x)|^p, B = |g(x)|^q, and θ = 1/p so that 1 - θ = 1/q, then (2) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Integrating this inequality yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the proof of the Hölder inequality is complete.

For the case when the equality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] holds, see Exercise 3.

We are now ready to prove the triangle inequality for the L^p norm.

Theorem 1.2 (Minkowski) If 1 ≤ p < ∞ and f, g [member of] L^p, then f + g [member of] L^p and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. The case p = 1 is obtained by integrating |f(x) + g(x)| ≤ |f(x)| + |g(x)|. When p > 1, we may begin by verifying that f + g [member of] L^p, when both f and g belong to L^p. Indeed,

|f(x) + g(x)|^p ≤ 2^p(|f(x)|^p + |g(x)|^p),

as can be seen by considering separately the cases |f(x)| ≤ |g(x)| and |g(x)| ≤ |f(x)|. Next we note that

|f(x) + g(x)|^p ≤ |f(x)| |f(x) + g(x)|^p-1 + |g(x)| |f(x) + g(x)|^p-1.

If q denotes the conjugate exponent of p, then (p - 1)q = p, so we see that (f + g)^p-1 belongs to L^q, and therefore Hölder's inequality applied to the two terms on the right-hand side of the above inequality gives

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, using once again (p - 1)q = p, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (3), since p - p/q = 1, and because we may suppose that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so the proof is finished.

1.2 Completeness of L^p

The triangle inequality makes L^p into a metric space with distance [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The basic analytic fact is that L^p is complete in the sense that every Cauchy sequence in the norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges to an element in L^p.

Taking limits is a necessity in many problems, and the L^p spaces would be of little use if they were not complete. Fortunately, like L¹ and L², the general L^p space does satisfy this desirable property.

Theorem 1.3 The space L^p(X,F, μ) is complete in the norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. The argument is essentially the same as for L¹ (or L²); see Section 2, Chapter 2 and Section 1, Chapter 4 in Book III. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a Cauchy sequence in L^p, and consider a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of {f_n} with the following property [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all k ≥ 1. We now consider the series whose convergence will be seen below

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the corresponding partial sums

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The triangle inequality for L^p implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Letting K tend to infinity, and applying the monotone convergence theorem proves that ∫ g^p < ∞, and therefore the series defining g, and hence the series defining f converges almost everywhere, and f [member of] L^p.

We now show that f is the desired limit of the sequence {f_n}. Since (by construction of the telescopic series) the (K - 1)^th partial sum of this series is precisely [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in L^p as well, we first observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all K. Then, we may apply the dominated convergence theorem to get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as K tends to infinity.

Finally, the last step of the proof consists of recalling that {f_n} is Cauchy. Given [member of] > 0, there exists N so that for all n, m > N we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If n_K is chosen so that n_K > N, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the triangle inequality implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever n > N. This concludes the proof of the theorem.

1.3 Further remarks

We begin by looking at some possible inclusion relations between the various L^p spaces. The matter is simple if the underlying space has finite measure.

Proposition 1.4 If X has finite positive measure, and p₀ ≤ p₁, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We may assume that p₁ > p₀. Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], G = 1, p = p₁/p₀ > 1, and 1/p + 1/q =1, in Hölder's inequality applied to F and G. This yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In particular, we find that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Moreover, by taking the p^th₀ root of both sides of the above equation, we find that the inequality in the proposition holds.

However, as is easily seen, such inclusion does not hold when X has infinite measure. (See Exercise 1). Yet, in an interesting special case the opposite inclusion does hold.

(Continues...)

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Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2011
• ISBN 10: 0691113874
• ISBN 13: 9780691113876
• Einband: Tapa dura
• Sprache: Englisch
• Anzahl der Seiten: 448
• Kontakt zum Hersteller: Nicht verfügbar