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E-grāmata: Real Analysis

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Real AnalysisPalielināt
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The second edition of this classic textbook presents a rigorous and self-contained introduction to real analysis with the goal of providing a solid foundation for future coursework and research in applied mathematics.  Written in a clear and concise style, it covers all of the necessary subjects as well as those often absent from standard introductory texts.  Each chapter features a Problems and Complements section that includes additional material that briefly expands on certain topics within the chapter and numerous exercises for practicing the key concepts.

The first eight chapters explore all of the basic topics for training in real analysis, beginning with a review of countable sets before moving on to detailed discussions of measure theory, Lebesgue integration, Banach spaces, functional analysis, and weakly differentiable functions.  More topical applications are discussed in the remaining chapters, such as maximal functions, functions of bounded mean oscillation, rearrangements, potential theory, and the theory of Sobolev functions.  This second edition has been completely revised and updated and contains a variety of new content and expanded coverage of key topics, such as new exercises on the calculus of distributions, a proof of the Riesz convolution, Steiner symmetrization, and embedding theorems for functions in Sobolev spaces.  

Ideal for either classroom use or self-study, Real Analysis is an excellent textbook both for students discovering real analysis for the first time and for mathematicians and researchers looking for a useful resource for reference or review.

Praise for the First Edition:

[ This book] will be extremely useful as a text.  There is certainly enough material for a year-long graduate course, but judicious selection would make it possible to use this most appealing book in a one-semester course for well-prepared students.  

Mathematical Reviews

Recenzijas

The book is a valuable, comprehensive reference source on real analysis. The first eight chapters cover core material that is part of most courses taught on the subject, followed by a collection of special topics that stay within the framework of real analysis. In addition to the content, what makes the book especially useful as a reference source is its organization. Summing Up: Recommended. Graduate students and faculty. This work should be used solely as a reference. (M. Bona, Choice, Vol. 54 (9), May, 2017)

The reader can find many interesting details which serve to illuminate the diamonds of analysis. The list of references contains the main books and articles which form the modern real analysis. The book can be recommended as one of the main readings on real analysis for those who are interested in this subject and its numerous applications. (Sergei V. Rogosin, zbMATH 1353.26001, 2017)

1 Preliminaries
1(16)
1 Countable Sets
1(1)
2 The Cantor Set
2(1)
3 Cardinality
3(2)
3.1 Some Examples
4(1)
4 Cardinality of Some Infinite Cartesian Products
5(2)
5 Orderings, the Maximal Principle, and the Axiom of Choice
7(1)
6 Well Ordering
8(9)
6.1 The First Uncountable
9(1)
Problems and Complements
10(1)
1c Countable Sets
10(1)
2c The Cantor Set
10(1)
2.1c A Generalized Cantor Set of Positive Measure
11(1)
2.2c A Generalized Cantor Set of Measure Zero
12(1)
2.3c Perfect Sets
13(1)
3c Cardinality
14(3)
2 Topologies and Metric Spaces
17(50)
1 Topological Spaces
17(2)
1.1 Hausdorff and Normal Spaces
18(1)
2 Urysohn's Lemma
19(1)
3 The Tietze Extension Theorem
20(2)
4 Bases, Axioms of Countability and Product Topologies
22(2)
4.1 Product Topologies
23(1)
5 Compact Topological Spaces
24(2)
5.1 Sequentially Compact Topological Spaces
25(1)
6 Compact Subsets of RN
26(2)
7 Continuous Functions on Countably Compact Spaces
28(1)
8 Products of Compact Spaces
28(2)
9 Vector Spaces
30(2)
9.1 Convex Sets
31(1)
9.2 Linear Maps and Isomorphisms
32(1)
10 Topological Vector Spaces
32(2)
10.1 Boundedness and Continuity
34(1)
11 Linear Functionals
34(2)
12 Finite Dimensional Topological Vector Spaces
36(1)
12.1 Locally Compact Spaces
36(1)
13 Metric Spaces
37(3)
13.1 Separation and Axioms of Countability
38(1)
13.2 Equivalent Metrics
39(1)
13.3 Pseudo Metrics
40(1)
14 Metric Vector Spaces
40(2)
14.1 Maps Between Metric Spaces
41(1)
15 Spaces of Continuous Functions
42(2)
15.1 Spaces of Continuously Differentiable Functions
43(1)
15.2 Spaces of Holder and Lipschitz Continuous Functions
44(1)
16 On the Structure of a Complete Metric Space
44(2)
16.1 The Uniform Boundedness Principle
45(1)
17 Compact and Totally Bounded Metric Spaces
46(21)
17.1 Pre-Compact Subsets of X
48(1)
Problems and Complements
48(1)
1c Topological Spaces
48(1)
1.12c Connected Spaces
49(1)
1.19c Separation Properties of Topological Spaces
49(1)
4c Bases, Axioms of Countability and Product Topologies
50(1)
4.10c The Box Topology
51(1)
5c Compact Topological Spaces
51(1)
5.8c The Alexandrov One-Point Compactification of {X;U} ([ 3])
52(1)
7c Continuous Functions on Countably Compact Spaces
53(1)
7.1c Upper-Lower Semi-continuous Functions
53(1)
7.2c Characterizing Lower-Semi Continuous Functions in RN
54(1)
7.3c On the Weierstrass-Baire Theorem
54(2)
7.4c On the Assumptions of Dini's Theorem
56(1)
9c Vector Spaces
56(1)
9.3c Hamel Bases
57(1)
9.6c On the Dimension of a Vector Space
58(1)
10c Topological Vector Spaces
58(1)
13c Metric Spaces
59(1)
13.10c The Hausdorff Distance of Sets
60(1)
13.11c Countable Products of Metric Spaces
61(1)
14c Metric Vector Spaces
62(1)
15c Spaces of Continuous Functions
63(1)
15.1c Spaces of Holder and Lipschitz Continuous Functions
63(1)
16c On the Structure of a Complete Metric Space
63(1)
16.3c Completion of a Metric Space
64(1)
16.4c Some Consequences of the Baire Category Theorem
65(1)
17c Compact and Totally Bounded Metric Spaces
65(1)
17.1c An Application of the Lebesgue Number Lemma
66(1)
3 Measuring Sets
67(66)
1 Partitioning Open Subsets of RN
67(1)
2 Limits of Sets, Characteristic Functions, and σ-Algebras
68(2)
3 Measures
70(3)
3.1 Finite, σ-Finite, and Complete Measures
72(1)
3.2 Some Examples
73(1)
4 Outer Measures and Sequential Coverings
73(2)
4.1 The Lebesgue Outer Measure in RN
74(1)
4.2 The Lebesgue--Stieltjes Outer Measure [ 89, 154]
75(1)
5 The Hausdorff Outer Measure in RN [ 71]
75(3)
5.1 Metric Outer Measures
77(1)
6 Constructing Measures from Outer Measures [ 26]
78(2)
7 The Lebesgue--Stieltjes Measure on R
80(1)
7.1 Borel Measures
81(1)
8 The Hausdorff Measure on RN
81(2)
9 Extending Measures from Semi-algebras to σ-Algebras
83(2)
9.1 On the Lebesgue--Stieltjes and Hausdorff Measures
85(1)
10 Necessary and Sufficient Conditions for Measurability
85(1)
11 More on Extensions from Semi-algebras to σ-Algebras
86(1)
12 The Lebesgue Measure of Sets in RN
87(3)
12.1 A Necessary and Sufficient Condition of Measurability
88(2)
13 Vitali's Nonmeasurable Set [ 168]
90(1)
14 Borel Sets, Measurable Sets, and Incomplete Measures
91(3)
14.1 A Continuous Increasing Function f: [ 0, 1] → [ 0, 1]
91(2)
14.2 On the Preimage of a Measurable Set
93(1)
14.3 Proof of Propositions 14.1 and 14.2
94(1)
15 Borel Measures
94(2)
16 Borel, Regular, and Radon Measures
96(2)
16.1 Regular Borel Measures
97(1)
16.2 Radon Measures
98(1)
17 Vitali Coverings
98(3)
18 The Besicovitch Covering Theorem
101(3)
19 Proof of Proposition 18.1
104(2)
20 The Besicovitch Measure-Theoretical Covering Theorem
106(27)
Problems and Complements
108(1)
1c Partitioning Open Subsets of RN
108(1)
2c Limits of Sets, Characteristic Functions and σ-Algebras
108(1)
3c Measures
109(2)
3.1c Completion of a Measure Space
111(1)
4c Outer Measures
112(1)
5c The Hausdorff Outer Measure in RN
112(1)
5.1c The Hausdorff Dimension of a Set E ⊂ RN
112(1)
5.2c The Hausdorff Dimension of the Cantor Set is ln 2/ln 3
113(1)
8c The Hausdorff Measure in RN
114(1)
8.1c Hausdorff Outer Measure of the Lipschitz Image of a Set
114(1)
8.2c Hausdorff Dimension of Graphs of Lipschitz Functions
115(1)
9c Extending Measures from Semi-algebras to σ-Algebras
115(1)
9.1c Inner and Outer Continuity of λ on Some Algebra Q
115(1)
10c More on Extensions from Semi-algebras to σ-Algebras
116(1)
10.1c Self-extensions of Measures
116(1)
10.2c Nonunique Extensions of Measures λ on Semi-algebras
117(1)
12c The Lebesgue Measure of Sets in RN
118(2)
12.1c Inner Measure and Measurability
120(1)
12.2c The Peano--Jordan Measure of Bounded Sets in RN
120(1)
12.3c Lipschitz Functions and Measurability
121(1)
13c Vitali's Nonmeasurable Set
122(1)
14c Borel Sets, Measurable Sets and Incomplete Measures
123(1)
16c Borel, Regular and Radon Measures
124(1)
16.1c Regular Borel Measures
124(1)
16.2c Regular Outer Measures
124(1)
17c Vitali Coverings
125(1)
17.1c Pointwise and Measure-Theoretical Vitali Coverings
125(1)
18c The Besicovitch Covering Theorem
126(1)
18.1c The Besicovitch Theorem for Unbounded E
126(1)
18.2c The Besicovitch Measure-Theoretical Inner Covering of Open Sets E ⊂ RN
127(1)
18.3c A Simpler Form of the Besicovitch Theorem
127(3)
18.4c Another Besicovitch-Type Covering
130(3)
4 The Lebesgue Integral
133(60)
1 Measurable Functions
133(2)
2 The Egorov--Severini Theorem [ 39, 145]
135(2)
2.1 The Egorov--Severini Theorem in RN
137(1)
3 Approximating Measurable Functions by Simple Functions
137(2)
4 Convergence in Measure (Riesz [ 125], Fisher [ 46])
139(2)
5 Quasicontinuous Functions and Lusin's Theorem
141(2)
6 Integral of Simple Functions ([ 87])
143(1)
7 The Lebesgue Integral of Nonnegative Functions
144(1)
8 Fatou's Lemma and the Monotone Convergence Theorem
145(2)
9 More on the Lebesgue Integral
147(2)
10 Convergence Theorems
149(1)
11 Absolute Continuity of the Integral
150(1)
12 Product of Measures
151(1)
13 On the Structure of (A × B)
152(3)
14 The Theorem of Fubini--Tonelli
155(2)
14.1 The Tonelli Version of the Fubini Theorem
156(1)
15 Some Applications of the Fubini--Tonelli Theorem
157(4)
15.1 Integrals in Terms of Distribution Functions
157(1)
15.2 Convolution Integrals
158(2)
15.3 The Marcinkiewicz Integral ([ 101, 102])
160(1)
16 Signed Measures and the Hahn Decomposition
161(2)
17 The Radon-Nikodym Theorem
163(4)
17.1 Sublevel Sets of a Measurable Function
164(1)
17.2 Proof of the Radon-Nikodym Theorem
165(2)
18 Decomposing Measures
167(26)
18.1 The Jordan Decomposition
168(1)
18.2 The Lebesgue Decomposition
168(1)
18.3 A General Version of the Radon-Nikodym Theorem
169(1)
Problems and Complements
170(1)
1c Measurable Functions
170(2)
1.1c Sublevel Sets
172(1)
2c The Egorov--Severini Theorem
173(1)
3c Approximating Measurable Functions by Simple Functions
173(1)
4c Convergence in Measure
174(1)
7c The Lebesgue Integral of Nonnegative Measurable Functions
175(1)
7.1c Comparing the Lebesgue Integral with the Peano-Jordan Integral
175(1)
7.2c On the Definition of the Lebesgue Integral
176(1)
9c More on the Lebesgue Integral
177(1)
10c Convergence Theorems
178(1)
10.1c Another Version of Dominated Convergence
178(3)
11c Absolute Continuity of the Integral
181(1)
12c Product of Measures
182(1)
12.1c Product of a Finite Sequence of Measure Spaces
182(1)
13c On the Structure of (A × B)
183(1)
13.1c Sections and Their Measure
184(1)
14c The Theorem of Fubini--Tonelli
185(1)
15c Some Applications of the Fubini--Tonelli Theorem
186(1)
15.1c Integral of a Function as the "Area Under the Graph"
186(1)
15.2c Distribution Functions
186(1)
17c The Radon-Nikodym Theorem
187(2)
18c A Proof of the Radon-Nikodym Theorem When Both μ and ν Are σ-Finite
189(4)
5 Topics on Measurable Functions of Real Variables
193(54)
1 Functions of Bounded Variation ([ 78])
193(2)
2 Dini Derivatives ([ 37])
195(2)
3 Differentiating Functions of Bounded Variation
197(1)
4 Differentiating Series of Monotone Functions
198(1)
5 Absolutely Continuous Functions ([ 91, 169])
199(2)
6 Density of a Measurable Set
201(1)
7 Derivatives of Integrals
202(2)
8 Differentiating Radon Measures
204(2)
9 Existence and Measurability of Dμν
206(2)
9.1 Proof of Proposition 9.2
208(1)
10 Representing Dμν
208(2)
10.1 Representing Dμν for ν << μ
208(2)
10.2 Representing Dμv for ν τ μ
210(1)
11 The Lebesgue-Besicovitch Differentiation Theorem
210(2)
11.1 Points of Density
211(1)
11.2 Lebesgue Points of an Integrable Function
211(1)
12 Regular Families
212(1)
13 Convex Functions
213(2)
14 The Jensen's Inequality
215(1)
15 Extending Continuous Functions
216(2)
15.1 The Concave Modulus of Continuity of f
216(2)
16 The Weierstrass Approximation Theorem
218(1)
17 The Stone-Weierstrass Theorem
219(1)
18 Proof of the Stone-Weierstrass Theorem
220(2)
18.1 Proof of Stone's Theorem
221(1)
19 The Ascoli-Arzela Theorem
222(25)
19.1 Pre-compact Subsets of C(E)
223(1)
Problems and Complements
224(1)
1c Functions of Bounded Variations
224(1)
1.1c The Function of The Jumps
225(1)
1.2c The Space BV[ a, b]
225(1)
2c Dini Derivatives
226(2)
2.1c A Continuous, Nowhere Differentiable Function ([ 167])
228(1)
2.2c An Application of the Baire Category Theorem
229(1)
4c Differentiating Series of Monotone Functions
229(1)
5c Absolutely Continuous Functions
229(1)
5.1c The Cantor Ternary Function ([ 23])
230(1)
5.2c A Continuous Strictly Monotone Function with a.e. Zero Derivative
231(2)
5.3c Absolute Continuity of the Distribution Function of a Measurable Function
233(1)
7c Derivatives of Integrals
234(2)
7.1c Characterizing BV[ a, b] Functions
236(2)
7.2c Functions of Bounded Variation in N Dimensions [ 55]
238(1)
13c Convex Functions
239(1)
13.8c Convex Functions in RN
240(1)
13.14c The Legendre Transform ([ 92])
241(1)
13.15c Finiteness and Coercivity
241(1)
13.16c The Young's Inequality
242(1)
14c Jensen's Inequality
243(1)
14.1c The Inequality of the Geometric and Arithmetic Mean
243(1)
14.2c Integrals and Their Reciprocals
243(1)
15c Extending Continuous Functions
244(1)
16c The Weierstrass Approximation Theorem
244(1)
17c The Stone-Weierstrass Theorem
244(1)
19c A General Version of the Ascoli-Arzela Theorem
245(2)
6 The LP Spaces
247(66)
1 Functions in Lp(E) and Their Norm
247(1)
2 The Holder and Minkowski Inequalities
248(2)
3 More on the Spaces Lp and Their Norm
250(2)
3.1 Characterizing the Norm ||f||p for 1 ≤ p < ∞
250(1)
3.2 The Norm || · ||∞ for E of Finite Measure
250(1)
3.3 The Continuous Version of the Minkowski Inequality
251(1)
4 Lp(E) for 1 ≤ p ≤ ∞ as Normed Spaces of Equivalence Classes
252(1)
4.1 Lp(E) for 1 ≤ p ≤ ∞ as a Metric Topological Vector Space
253(1)
5 Convergence in Lp(E) and Completeness
253(2)
6 Separating Lp(E) by Simple Functions
255(2)
7 Weak Convergence in Lp(E)
257(1)
7.1 Counterexample
257(1)
8 Weak Lower Semi-continuity of the Norm in Lp(E)
258(1)
9 Weak Convergence and Norm Convergence
259(2)
9.1 Proof of Proposition 9.1 for p ≥ 2
260(1)
9.2 Proof of Proposition 9.1 for 1 < p < 2
261(1)
10 Linear Functionals in Lp(E)
261(2)
11 The Riesz Representation Theorem
263(3)
11.1 Proof of Theorem 11.1: The Case of {X, A, μ} Finite
263(1)
11.2 Proof of Theorem 11.1: The Case of {X, A, μ} σ-Finite
264(1)
11.3 Proof of Theorem 11.1: The Case 1 < p < ∞
265(1)
12 The Hanner and Clarkson Inequalities
266(3)
12.1 Proof of Hanner's Inequalities
268(1)
12.2 Proof of Clarkson's Inequalities
268(1)
13 Uniform Convexity of Lp(E) for 1 < p < ∞
269(2)
14 The Riesz Representation Theorem By Uniform Convexity
271(2)
14.1 Proof of Theorem 14.1. The Case 1 < p < ∞
271(1)
14.2 The Case p = 1 and E of Finite Measure
272(1)
14.3 The Case p = 1 and {X, A, μ} σ-Finite
273(1)
15 If E ⊂ RN and p ε [ 1, ∞), then Lp(E) Is Separable
273(3)
15.1 L∞{E) Is Not Separable
276(1)
16 Selecting Weakly Convergent Subsequences
276(1)
17 Continuity of the Translation in Lp(E) for 1 ≤ p < ∞
277(3)
17.1 Continuity of the Convolution
280(1)
18 Approximating Functions in Lp(E) with Functions in C∞(E)
280(3)
19 Characterizing Pre-compact Sets in Lp(E)
283(30)
Problems and Complements
285(1)
1c Functions in Lp(E) and Their Norm
285(1)
1.1c The Spaces Lp for 0 < p < 1
285(1)
1.2c The Spaces Lp for q < 0
285(1)
1.3c The Spaces lp for 1 ≤ p ≤ ∞
285(1)
2c The Inequalities of Holder and Minkowski
286(1)
2.1c Variants of the Holder and Minkowski Inequalities
286(1)
2.2c Some Auxiliary Inequalities
287(1)
2.3c An Application to Convolution Integrals
287(1)
2.4c The Reverse Holder and Minkowski Inequalities
288(1)
2.5c Lp(E)-Norms and Their Reciprocals
288(1)
3c More on the Spaces Lp and Their Norm
288(1)
3.4c A Metric Topology for Lp(E) when 0 < p < 1
289(1)
3.5c Open Convex Subsets of Lp(E) for 0 < p < 1
289(1)
5c Convergence in Lp(E) and Completeness
290(1)
5.1c The Measure Space {X, A, μ} and the Metric Space {A; d}
291(1)
6c Separating Lp{E) by Simple Functions
291(1)
7c Weak Convergence in Lp{E)
292(1)
7.3c Comparing the Various Notions of Convergence
293(1)
7.5c Weak Convergence in lp
294(1)
9c Weak Convergence and Norm Convergence
295(1)
9.1c Proof of Lemmas 9.1 and 9.2
295(1)
11c The Riesz Representation Theorem
296(1)
11.1c Weakly Cauchy Sequences in Lp(X) for 1 < P ≤ ∞
296(1)
11.2c Weakly Cauchy Sequences in Lp(X) for p = 1
297(1)
11.3c The Riesz Representation Theorem in lp
297(1)
14c The Riesz Representation Theorem By Uniform Convexity
297(1)
14.1c Bounded Linear Functional in Lp(E) for 0 < p < 1
297(1)
14.2c An Alternate Proof of Proposition 14.1c
298(1)
15c If E ⊂ RN and p ε [ 1, ∞), then Lp(E) Is Separable
299(1)
18c Approximating Functions in Lp(E) with Functions in C∞(E)
299(1)
18.1c Caloric Extensions of Functions in Lp(RN)
299(2)
18.2c Harmonic Extensions of Functions in Lp(RN)
301(2)
18.3c Characterizing Holder Continuous Functions
303(1)
19c Characterizing Pre-compact Sets in Lp(E)
304(1)
19.1c The Helly's Selection Principle
304(1)
20c The Vitali-Saks-Hahn Theorem [ 59, 138, 170]
305(2)
21c Uniformly Integrable Sequences of Functions
307(2)
22c Relating Weak and Strong Convergence and Convergence in Measure
309(2)
23c An Independent Proof of Corollary 22.1c
311(2)
7 Banach Spaces
313(66)
1 Normed Spaces
313(2)
1.1 Semi-norms and Quotients
314(1)
2 Finite and Infinite Dimensional Normed Spaces
315(3)
2.1 A Counterexample
315(1)
2.2 The Riesz Lemma
316(1)
2.3 Finite Dimensional Spaces
317(1)
3 Linear Maps and Functionals
318(2)
4 Examples of Maps and Functionals
320(1)
4.1 Functionals
320(1)
4.2 Linear Functionals on C(E)
321(1)
4.3 Linear Functionals on C(E) for Some α ε (0, 1)
321(1)
5 Kernels of Maps ad Functionals
321(1)
6 Equibounded Families of Linear Maps
322(2)
6.1 Another Proof of Proposition 6.1
323(1)
7 Contraction Mappings
324(1)
7.1 Applications to Some Fredholm Integral Equations
325(1)
8 The Open Mapping Theorem
325(3)
8.1 Some Applications
327(1)
8.2 The Closed Graph Theorem
327(1)
9 The Hahn--Banach Theorem
328(2)
10 Some Consequences of the Hahn--Banach Theorem
330(2)
10.1 Tangent Planes
332(1)
11 Separating Convex Subsets of a Hausdorff, Topological Vector Space {X;U}
332(3)
11.1 Separation in Locally Convex, Hausdorff, Topological Vector Spaces {X;U}
334(1)
12 Weak Topologies
335(3)
12.1 Weak Boundedness
336(1)
12.2 Weakly and Strongly Closed Convex Sets
337(1)
13 Reflexive Banach Spaces
338(2)
14 Weak Compactness
340(2)
14.1 Weak Sequential Compactness
340(2)
15 The Weak* Topology of X*
342(1)
16 The Alaoglu Theorem
343(2)
17 Hilbert Spaces
345(1)
17.1 The Schwarz Inequality
345(1)
17.2 The Parallelogram Identity
346(1)
18 Orthogonal Sets, Representations and Functionals
346(3)
18.1 Bounded Linear Functionals on H
348(1)
19 Orthonormal Systems
349(2)
19.1 The Bessel Inequality
349(1)
19.2 Separable Hilbert Spaces
350(1)
20 Complete Orthonormal Systems
351(28)
20.1 Equivalent Notions of Complete Systems
352(1)
20.2 Maximal and Complete Orthonormal Systems
352(1)
20.3 The Gram--Schmidt Orthonormalization Process ([ 142])
352(1)
20.4 On the Dimension of a Separable Hilbert Space
353(1)
Problems and Complements
353(1)
1c Normed Spaces
353(1)
1.1c Semi-Norms and Quotients
354(1)
2c Finite and Infinite Dimensional Normed Spaces
355(1)
3c Linear Maps and Functionals
356(1)
6c Equibounded Families of Linear Maps
357(1)
8c The Open Mapping Theorem
358(1)
9c The Hahn--Banach Theorem
358(1)
9.1c The Complex Hahn--Banach Theorem
359(1)
9.2c Linear Functionals in L∞(E)
359(1)
11c Separating Convex Subsets of X
360(1)
11.1c A Counterexample of Tukey [ 164]
360(1)
11.2c A Counterexample of Goffman and Pedrick [ 56]
361(1)
11.3c Extreme Points of a Convex Set
361(2)
11.4c A General Version of the Krein--Milman Theorem
363(1)
12c Weak Topologies
363(1)
12.1c Infinite Dimensional Normed Spaces
364(1)
12.2c About Corollary 12.5
365(1)
12.3c Weak Closure and Weak Sequential Closure
365(3)
14c Weak Compactness
368(1)
14.1c Linear Functionals on Subspaces of C(E)
368(1)
14.2c Weak Compactness and Boundedness
369(1)
15c The Weak* Topology of X*
369(1)
15.1c Total Sets of X
369(1)
15.2c Metrization Properties of Weak* Compact Subsets of X*
370(1)
16c The Alaoglu Theorem
371(1)
16.1c The Weak* Topology of X**
372(2)
16.2c Characterizing Reflexive Banach Spaces
374(1)
16.3c Metrization Properties of the Weak Topology of the Closed Unit Ball of a Banach Space
374(1)
16.4c Separating Closed Sets in a Reflexive Banach Space
375(1)
17c Hilbert Spaces
376(1)
17.1c On the Parallelogram Identity
376(1)
18c Orthogonal Sets, Representations and Functionals
376(1)
19c Orthonormal Systems
377(2)
8 Spaces of Continuous Functions, Distributions, and Weak Derivatives
379(52)
1 Bounded Linear Functionals on Co(RN)
379(2)
1.1 Positive Linear Functionals on Co(RN)
380(1)
1.2 The Riesz Representation Theorem
380(1)
2 Partition of Unity
381(1)
3 Proof of Theorem 1.1. Constructing μ
381(2)
4 An Auxiliary Positive Linear Functional on Co(RN)+
383(2)
4.1 Measuring Compact Sets by T+
384(1)
5 Representing T+ on Co(RN)+ as in (1.1) for a Unique μB
385(1)
6 Proof of Theorem 1.1. Representing T on Co(RN) as in (1.3) for a Unique μ-Measurable w
386(1)
7 A Topology for C∞o(E) for an Open Set E ⊂ RN
387(2)
8 A Metric Topology for C∞o(E)
389(2)
8.1 Equivalence of These Topologies
389(1)
8.2 D(E) Is Not Complete
390(1)
9 A Topology for C∞o(K) for a Compact Set K ⊂ E
391(1)
9.1 D(K) Is Complete
391(1)
9.2 Relating the Topology of D(E) to the Topology of D(K)
392(1)
10 The Schwartz Topology of D(E)
392(1)
11 D(E) Is Complete
393(2)
11.1 Cauchy Sequences in D(E) and Completeness
394(1)
11.2 The Topology of D(E) Is Not Metrizable
394(1)
12 Continuous Maps and Functionals
395(1)
12.1 Distributions in E
395(1)
12.2 Continuous Linear Maps T: D(E) → D(E)
396(1)
13 Distributional Derivatives
396(2)
14 Fundamental Solutions
398(3)
14.1 The Fundamental Solution of the Wave Operator in R2
399(1)
14.2 The Fundamental Solution of the Laplace Operator
400(1)
15 Weak Derivatives and Main Properties
401(2)
16 Domains and Their Boundaries
403(2)
16.1 ∂E of Class C1
403(1)
16.2 Positive Geometric Density and ∂E Piecewise Smooth
404(1)
16.3 The Segment Property
404(1)
16.4 The Cone Property
405(1)
16.5 On the Various Properties of ∂E
405(1)
17 More on Smooth Approximations
405(2)
18 Extensions into RN
407(2)
19 The Chain Rule
409(1)
20 Steklov Averagings
410(3)
20.1 Characterizing W1,p(E) for 1 < p < ∞
412(1)
20.2 Remarks on W1,∞(E)
413(1)
21 The Rademacher's Theorem
413(18)
Problems and Complements
413(2)
1c Bounded Linear Functionals on Co(RN; Rm)
415(1)
2c Convergence of Measures
416(2)
3c Calculus with Distributions
418(3)
4c Limits in D'
421(3)
5c Algebraic Equations in D'
424(2)
6c Differential Equations in D'
426(2)
7c Miscellaneous Problems
428(3)
9 Topics on Integrable Functions of Real Variables
431(60)
1 A Vitali-Type Covering
431(2)
2 The Maximal Function (Hardy--Littlewood [ 69] and Wiener [ 175])
433(2)
3 Strong Lp Estimates for the Maximal Function
435(2)
3.1 Estimates of Weak and Strong Type
436(1)
4 The Calderon--Zygmund Decomposition Theorem [ 20]
437(1)
5 Functions of Bounded Mean Oscillation
438(3)
5.1 Some Consequences of the John--Nirenberg Theorem
439(2)
6 Proof of the John--Nirenberg Theorem 5.1
441(3)
7 The Sharp Maximal Function
444(1)
8 Proof of the Fefferman--Stein Theorem
445(2)
9 The Marcinkiewicz Interpolation Theorem
447(2)
9.1 Quasi-linear Maps and Interpolation
448(1)
10 Proof of the Marcinkiewicz Theorem
449(2)
11 Rearranging the Values of a Function
451(2)
12 Some Integral Inequalities for Rearrangements
453(4)
12.1 Contracting Properties of Symmetric Rearrangements
454(1)
12.2 Testing for Measurable Sets E Such that = E* a.e. in RN
455(2)
13 The Riesz Rearrangement Inequality
457(1)
13.1 Reduction to Characteristic Functions of Bounded Sets
457(1)
14 Proof of (13.1) for N = 1
458(5)
14.1 Reduction to Finite Union of Intervals
458(2)
14.2 Proof of (13.1) for N =
1. The Case T + S ≤ R
460(1)
14.3 Proof of (13.1) for N =
1. The Case S + T > R
461(2)
14.4 Proof of the Lemma 14.1
463(1)
15 The Hardy's Inequality
463(2)
16 The Hardy--Littlewood--Sobolev Inequality for N = 1
465(1)
16.1 Some Reductions
466(1)
17 Proof of Theorem 16.1
466(2)
18 The Hardy--Littlewood--Sobolev Inequality for N ≥ 1
468(1)
18.1 Proof of Theorem 18.1
468(1)
19 Potential Estimates
469(1)
20 Lp Estimates of Riesz Potentials
470(2)
20.1 Motivating Lp Estimates of Riesz Potentials as Embeddings
471(1)
21 Lp Estimates of Riesz Potentials for p = 1 and p > N
472(1)
22 The Limiting Case p = N
473(2)
23 Steiner Symmetrization of a Set E ⊂ RN
475(2)
24 Some Consequences of Steiner's Symmetrization
477(2)
24.1 Symmetrizing a Set About the Origin
477(1)
24.2 The Isodiametric Inequality
478(1)
24.3 Steiner Rearrangement of a Function
479(1)
25 Proof of the Riesz Rearrangement Inequality for N = 2
479(5)
25.1 The Limit of {Fn}
481(2)
25.2 The Set F* Is the Disc F*
483(1)
26 Proof of the Riesz Rearrangement Inequality for N > 2
484(7)
Problems and Complements
485(1)
11c Rearranging the Values of a Function
485(1)
12c Some Integral Inequalities for Rearrangements
486(1)
20c Lp Estimates of Riesz Potentials
486(1)
21c Lp Estimates of Riesz Potentials for p = 1 and p ≥ N
487(1)
22c The Limiting Case p = N/α
487(1)
23c Some Consequences of Steiner's Symmetrization
488(1)
23.1c Applications of the Isodiametric Inequality
488(3)
10 Embeddings of W1,p(E) into Lq(E)
491(50)
1 Multiplicative Embeddings of W1,po(E)
491(2)
1.1 Proof of Theorem 1.1
493(1)
2 Proof of Theorem 1.1 for N = 1
493(1)
3 Proof of Theorem 1.1 for 1 ≤ p < N
494(2)
4 Proof of Theorem 1.1 for 1 ≤ p < N Concluded
496(1)
5 Proof of Theorem 1.1 for p ≥ N > 1
496(2)
5.1 Estimate of I1(x, R)
497(1)
5.2 Estimate of I2(x, R)
498(1)
6 Proof of Theorem 1.1 for p ≥ N > 1 Concluded
498(1)
7 On the Limiting Case p = N
499(1)
8 Embeddings of W1,p(E)
500(1)
9 Proof of Theorem 8.1
501(2)
10 Poincare Inequalities
503(4)
10.1 The Poincare Inequality
504(1)
10.2 Multiplicative Poincare Inequalities
505(1)
10.3 Extensions of (u -- uE) for Convex E
506(1)
11 Level Sets Inequalities
507(1)
12 Morrey Spaces [ 110]
508(2)
12.1 Embeddings for Functions in the Morrey Spaces
509(1)
13 Limiting Embedding of W1,N(E)
510(2)
14 Compact Embeddings
512(2)
15 Fractional Sobolev Spaces in RN
514(2)
16 Traces
516(2)
17 Traces and Fractional Sobolev Spaces
518(1)
18 Traces on ∂E of Functions in W1,p(E)
519(3)
18.1 Traces and Fractional Sobolev Spaces
521(1)
19 Multiplicative Embeddings of W1,p(E)
522(2)
20 Proof of Theorem 19.1. A Special Case
524(2)
21 Constructing a Map Between E and Q. Part I
526(2)
21.1 Case
1. Σn Intersects Bρ
527(1)
21.2 Case
2. Σn Does not Intersect Bρ
528(1)
22 Constructing a Map Between E and Q. Part II
528(2)
23 Proof of Theorem 19.1 Concluded
530(1)
24 The Spaces W1,pp*(E)
530(11)
Problems and Complements
534(1)
1c Multiplicative Embeddings of W1,po(E)
534(1)
8c Embeddings of W1,p(E)
534(1)
8.1c Differentiability of Functions in W1,p(E) for p ≥ N
535(1)
14c Compact Embeddings
536(1)
17c Traces and Fractional Sobolev Spaces
536(1)
17.1c Characterizing Functions in W1-1/p,p(RN) as Traces
536(2)
18c Traces on ∂E of Functions in W1,p(E)
538(1)
18.1c Traces on a Sphere
538(3)
11 Topics on Weakly Differentiable Functions
541(38)
1 Sard's Lemma [ 140]
541(3)
2 The Co-area Formula for Smooth Functions
544(1)
3 The Isoperimetric Inequality for Bounded Sets E with Smooth Boundary ∂E
545(2)
3.1 Embeddings of W1,po(E) Versus the Isoperimetric Inequality
546(1)
4 The p-Capacity of a Compact Set K ⊂ RN, for 1 ≤ P < N
547(3)
4.1 Enlarging the Class of Competing Functions
548(2)
5 A Characterization of the p-Capacity of a Compact Set K ⊂ RN, for 1 ≤ P < N
550(2)
6 Lower Estimates of cp(K) for 1 ≤ P < N
552(3)
6.1 A Simpler Proof of Lemma 6.1 with a Coarser Constant
553(1)
6.2 p-Capacity of a Closed Ball Bρ ⊂ RN, for 1 ≤ p < N
554(1)
6.3 cp(Bρ) = cp(∂Bρ)
555(1)
7 The Norm ||Du||p, for 1 ≤ P < N, in Terms of the p-Capacity Distribution Function of u ε C∞o(RN)
555(3)
7.1 Some Auxiliary Estimates for 1 < p < N
555(2)
7.2 Proof of Theorem 7.1
557(1)
8 Relating Gagliardo Embeddings, Capacities, and the Isoperimetric Inequality
558(1)
9 Relating HN--p(K) to cp(K) for 1 < p < N
559(3)
9.1 An Auxiliary Proposition
559(2)
9.2 Proof of Theorem 9.1
561(1)
10 Relating cp(K) to HN--p+ε(K) for 1 ≤ p < N
562(1)
11 The p-Capacity of a Set E ⊂ RN for 1 ≤ P < N
563(3)
12 Limits of Sets and Their Outer p-Capacities
566(1)
13 Capacitable Sets
567(3)
14 Capacities Revisited and p-Capacitability of Borel Sets
570(2)
14.1 The Borel Sets in RN Are p-Capacitable
571(1)
14.2 Generating Measures by p-Capacities
572(1)
15 Precise Representatives of Functions in L1loc(RN)
572(2)
16 Estimating the p-Capacity of [ u > t] for t > 0
574(2)
17 Precise Representatives of Functions in W1,ploc(RN)
576(3)
17.1 Quasi-Continuous Representatives of Functions u ε W1,ploc(RN)
578(1)
References 579(6)
Index 585
Emmanuele DiBenedetto is Centennial Professor of Mathematics at Vanderbilt University, Nashville, TN, USA.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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