## Real AnalysisThe 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 |

473 | |

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Spectral Theory of Schrödinger Operators Rafael del Río,Carlos Villegas-Blas No preview available - 2004 |