Real Analysis

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Springer Science & Business Media, Apr 19, 2002 - Mathematics - 485 pages
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The focus of this modern graduate text in real analysis is to prepare the potential researcher to a rigorous "way of thinking" in applied mathematics and partial differential equations. The book will provide excellent foundations and serve as a solid building block for research in analysis, PDEs, the calculus of variations, probability, and approximation theory. All the core topics of the subject are covered, from a basic introduction to functional analysis, to measure theory, integration and weak differentiation of functions, and in a presentation that is hands-on, with little or no unnecessary abstractions.

Additional features:

* Carefully chosen topics, some not touched upon elsewhere: fine properties of integrable functions as they arise in applied mathematics and PDEs – Radon measures, the Lebesgue Theorem for general Radon measures, the Besicovitch covering Theorem, the Rademacher Theorem; topics in Marcinkiewicz integrals, functions of bounded variation, Legendre transform and the characterization of compact subset of some metric function spaces and in particular of Lp spaces

* Constructive presentation of the Stone-Weierstrass Theorem

* More specialized chapters (8-10) cover topics often absent from classical introductiory texts in analysis: maximal functions and weak Lp spaces, the Calderón-Zygmund decomposition, functions of bounded mean oscillation, the Stein-Fefferman Theorem, the Marcinkiewicz Interpolation Theorem, potential theory, rearrangements, estimations of Riesz potentials including limiting cases

* Provides a self-sufficient introduction to Sobolev Spaces, Morrey Spaces and Poincaré inequalities as the backbone of PDEs and as an essential environment to develop modern and current analysis

* Comprehensive index

This clear, user-friendly exposition of real analysis covers a great deal of territory in a concise fashion, with sufficient motivation and examples throughout. A number of excellent problems, as well as some remarkable features of the exercises, occur at the end of every chapter, which point to additional theorems and results. Stimulating open problems are proposed to engage students in the classroom or in a self-study setting.

 

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Contents

Preliminaries
1
2 The Cantor set
2
3 Cardinality
4
31 Some examples
5
4 Cardinality of some infinite Cartesian products
6
5 Orderings the maximal principle and the axiom of choice
8
6 Wellordering
9
61 The first uncountable
11
16 The Riesz representation theorem by uniform convexity
247
162 The case where p 1 and E is of finite measure
248
163 The case where p 1 and X 𝓐 μ is 𝛿finite
249
17 Bounded linear functional in LPE for 0 p 1
250
18 If E ℝN and p ϵ 1 then LPE is separable
251
181 LE is not separable
254
20 Continuity of the translation in LPE for 1 p
255
21 Approximating functions in LPE with functions in CE
257

Topologies and Metric Spaces
17
11 Hausdorff and normal spaces
19
3 The Tietze extension theorem
21
4 Bases axioms of countability and product topologies
22
41 Product topologies
24
5 Compact topological spaces
25
51 Sequentially compact topological spaces
26
6 Compact subsets of Rⁿ
27
7 Continuous functions on countably compact spaces
29
8 Products of compact spaces
30
9 Vector spaces
31
91 Convex sets
33
10 Topological vector spaces
34
101 Boundedness and continuity
35
11 Linear functionals
36
121 Locally compact spaces
37
13 Metric spaces
38
131 Separation and axioms of countability
39
132 Equivalent metrics
40
14 Metric vector spaces
41
141 Maps between metric spaces
42
15 Spaces of continuous functions
43
151 Spaces of continuously differentiable functions
44
17 Compact and totally bounded metric spaces
46
171 Precompact subsets of X
48
Problems and Complements
49
Measuring Sets
65
2 Limits of sets characteristic functions and 𝛿algebras
67
3 Measures
68
31 Finite 𝛿finite and complete measures
71
4 Outer measures and sequential coverings
72
41 The Lebesgue outer measure in ℝN
73
5 The Hausdorff outer measure in ℝN
74
6 Constructing measures from outer measures
76
7 The LebesgueStieltjes measure on ℝ
79
71 Borel measures
80
9 Extending measures from semialgebras to 𝛿algebras
82
91 On the LebesgueStieltjes and Hausdorff measures
84
11 More on extensions from semialgebras to a 𝛿algebras
86
12 The Lebesgue measure of sets in RN
88
13 A nonmeasurable set
90
14 Borel sets measurable sets and incomplete measures
91
142 On the preimage of a measurable set
93
143 Proof of Propositions 141 and 142
94
151 Some extensions to general Borel measures
97
16 Regular outer measures and Radon measures
98
161 More on Radon measures
99
18 The Besicovitch covering theorem
103
19 Proof of Proposition 182
105
20 The Besicovitch measuretheoretical covering theorem
107
Problems and Complements
110
The Lebesgue Integral
123
2 The Egorov theorem
126
21 The Egorov theorem in RN
128
4 Convergence in measure
130
5 Quasicontinuous functions and Lusins theorem
133
6 Integral of simple functions
135
7 The Lebesgue integral of nonnegative functions
136
8 Fatous lemma and the monotone convergence theorem
137
9 Basic properties of the Lebesgue integral
139
10 Convergence theorems
141
11 Absolute continuity of the integral
142
13 On the structure of A x B
144
14 The FubiniTonelli theorem
147
141 The Tonelli version of the Fubini theorem
148
152 Convolution integrals
149
153 The Marcinkiewicz integral
150
16 Signed measures and the Hahn decomposition
151
17 The RadonNikodym theorem
154
18 Decomposing measures
157
182 The Lebesgue decomposition
159
183 A general version of the RadonNikodym theorem
160
Topics on Measurable Functions of Real Variables
171
2 Dini derivatives
173
3 Differentiating functions of bounded variation
176
4 Differentiating series of monotone functions
177
5 Absolutely continuous functions
179
6 Density of a measurable set
181
7 Derivatives of integrals
182
8 Differentiating Radon measures
184
9 Existence and measurability of D𝝁𝝂
186
91 Proof of Proposition 92
188
10 Representing D𝝁𝝂
189
102 Representing D𝝁𝝂 for 𝝂 𝝁
191
111 Points of density
192
12 Regular families
193
13 Convex functions
194
14 Jensens inequality
196
15 Extending continuous functions
197
16 The Weierstrass approximation theorem
199
17 The StoneWeierstrass theorem
200
18 Proof of the StoneWeierstrass theorem
201
181 Proof of Stones theorem
202
19 The AscoliArzela theorem
203
191 Precompact subsets of CE
204
Problems and Complements
205
The LPE Spaces
221
11 The spaces Lp for 0 p 1
222
2 The Hölder and Minkowski inequalities
223
3 The reverse Hölder and Minkowski inequalities
224
4 More on the spaces Lp and their norms
225
42 The norm for E of finite measure
226
43 The continuous version of the Minkowski inequality
227
51 LPE for 1 p as a metric topnlogical vector space
228
6 A metric topology for LPE when 0 p 1
229
7 Convergence in LPE and completeness
230
8 Separating LPE by simple functions
232
9 Weak convergence in LPE
234
10 Weak lower semicontinuity of the norm in LPE
235
11 Weak convergence and norm convergence
236
111 Proof of Proposition 111 for p 2
237
12 Linear functionals in LPE
238
13 The Riesz representation theorem
239
The case where XA𝝁 is finite
240
The case where XA𝝁 is 𝛿finite
241
The case where 1 p
242
14 The Hanner and Clarkson inequalities
243
141 Proof of Hanners inequalities
244
142 Proof of Clarksons inequalities
245
15 Uniform convexity of LPE for 1 p
246
22 Characterizing precompact sets in LPE
260
Problems and Complements
262
Banach Spaces
275
11 Seminorms and quotients
276
2 Finite and infinitedimensional normed spaces
277
22 The Riesz lemma
278
23 Finitedimensional spaces
279
3 Linear maps and functionals
280
4 Examples of maps and functionals
282
41 Functionals
283
5 Kernels of maps and functionals
284
6 Equibounded families of linear maps
285
61 Another proof of Proposition 61
286
71 Applications to some Fredholm integral equations
287
8 The open mapping theorem
288
81 Some applications
289
9 The HahnBanach theorem
290
10 Some consequences of the HahnBanach theorem
292
101 Tangent planes
295
12 Weak topologies
297
121 Weakly and strongly closed convex sets
299
13 Reflexive Banach spaces
300
14 Weak compactness
301
141 Weak sequential compactness
302
15 The weak topology
303
16 The Alaoglu theorem
304
17 Hilbert spaces
306
171 The Schwarz inequality
307
18 Orthogonal sets representations and functionals
308
181 Rounded linear functionals on H
310
191 The Bessel inequality
311
192 Separable Hilbert spaces
312
201 Equivalent notions of complete systems
313
203 The GramSchmidt orthonormalization process
314
Spaces of Continuous Functions Distributions and Weak Derivatives
325
11 Partition of unity
326
2 Bounded linear functionals on CℴℝN
327
22 Characterizing C0KN
328
Constructing the measure μ
331
Representing T as in 33
333
6 Characterizing bounded linear functionals on C0RN
335
62 Bounded linear functionals on C0RN
336
7 A topology for CE for an open set E RN
337
8 A metric topology for CE
339
81 Equivalence of these topologies
340
82 DE is not complete
341
91 A metric topology for CK
342
10 Relating the topology of DE to the topology of DK
343
101 Noncompleteness of DE
344
12 DE is complete
346
121 Cauchy sequences in DE
347
13 Continuous maps and functionals
348
DE DE
349
141 Derivatives of distributions
350
143 Miscellaneous remarks
351
15 Fundamental Solutions
352
152 The fundamental solution of the Laplace operator
354
16 Weak derivatives and main properties
355
17 Domains and their boundaries
358
174 The cone property
359
19 Extensions into RN
361
20 The chain rule
363
21 Steklov averagings
365
22 Characterizing W¹PE for 1 p
367
221 Remarks on W¹pE
368
Problems and Complements
371
Topics on Integrable Functions of Real Variables
375
2 The maximal function
377
3 Strong Lp estimates for the maximal function
379
31 Estimates of weak and strong type
380
4 The CalderonZygmund decomposition theorem
381
5 Functions of bounded mean oscillation
383
6 Proof of Theorem 51
384
7 The sharp maximal function
387
8 Proof of the FeffermanStein theorem
388
9 The Marcinkiewicz interpolation theorem
390
91 Quasilinear maps and interpolation
391
10 Proof of the Marcinkiewicz theorem
392
11 Rearranging the values of a function
394
12 Basic properties of rearrangements
396
13 Symmetric rearrangements
398
14 A convolution inequality for rearrangements
400
15 Reduction to a finite union of intervals
402
The case where T + S R
404
171 Proof of Lemma 171
407
19 A convolutiontype inequality
409
20 Proof of Theorem 191
410
21 An equivalent form of Theorem 191
411
22 An Ndimensional version of Theorem 211
412
23 Lp estimates of Riesz potentials
413
24 The limiting case p N
415
Problems and Complements
417
Embeddings of W¹pE into LqE
423
2 Proof of Theorem 11 for N 1
425
4 Proof of Theorem 11 for 1 p N concluded
428
51 Estimate of I₁x R
429
52 Estimate of 1₂x R For all v 6 RN
430
7 On the limiting case p N
431
8 Embeddings of W1PE
432
9 Proof of Theorem 81
433
10 Poincaré inequalities
435
102 Multiplicative Poincaré inequalities
437
11 The discrete isoperimetric inequality
438
12 Morrey spaces
439
121 Emheddings for functions in the Morrey spaces
440
13 Limiting embedding of W¹lNE
441
14 Compact embeddings
443
15 Fractional Sobolev spaces in ℝN
445
16 Traces
447
17 Traces and fractional Sobolev spaces
448
18 Traces on ǝE of functions in W¹pE
450
181 Traces and fractional Sobolev spaces
453
A special case
456
Part 1
458
Part 2
460
23 Proof of Theorem 191 concluded
463
Problems and Complements
464
References
469
Index
473
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