Verfügbarkeit: Real analysis

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Bibliographische Detailangaben
Titel:	Real analysis
Von:	Emmanuele DiBenedetto
Person:	DiBenedetto, Emmanuele 1947-2021 Verfasser aut
Hauptverfasser:	DiBenedetto, Emmanuele 1947-2021 (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Boston [u.a.] Birkhäuser 2002
Schriftenreihe:	Birkhäuser advanced texts
Schlagworte:	Analyse mathématique Análisis matemático Mathematical analysis Reelle Analysis
Online-Zugang:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009739572&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Beschreibung:	XXIV, 485 S.
ISBN:	0817642315 3764342315

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adam_text	Contents Preface xv Acknowledgments xxiii Preliminaries 1 1 Countable sets ............................ 1 2 The Cantor set ............................ 2 3 Cardinality .............................. 4 3.1 Some examples ......................... 5 4 Cardinality of some infinite Cartesian products ........... 6 5 Orderings, the maximal principle, and the axiom of choice ..... 8 6 Well-ordering ............................. 9 6.1 The first uncountable ...................... 11 Problems and Complements ....................... 11 I Topologies and Metric Spaces 17 1 Topological spaces .......................... 17 1.1 Hausdorff and normal spaces .................. 19 2 Urysohn s lemma ........................... 19 3 The Tietze extension theorem .................... 21 4 Bases, axioms of countability, and product topologies ........ 22 4.1 Product topologies ....................... 24 5 Compact topological spaces ..................... 25 5.1 Sequentially compact topological spaces ............ 26 vi Contents 6 Compact subsets of RN ....................... 27 7 Continuous functions on countably compact spaces ......... 29 8 Products of compact spaces ..................... 30 9 Vector spaces ............................. 31 9.1 Convex sets ........................... 33 9.2 Linear maps and isomorphisms ................ 33 10 Topological vector spaces ...................... 34 10.1 Boundedness and continuity .................. 35 1 1 Linear functionate .......................... 36 12 Finite-dimensional topological vector spaces ............ 36 12.1 Locally compact spaces .................... 37 13 Metric spaces ............................. 38 13.1 Separation and axioms of countability ............. 39 13.2 Equivalent metrics ....................... 40 13.3 Pseudometrics ......................... 40 14 Metric vector spaces ......................... 41 14.1 Maps between metric spaces .................. 42 15 Spaces of continuous functions .................... 43 15.1 Spaces of continuously differentiable functions ........ 44 16 On the structure of a complete metric space ............. 44 17 Compact and totally bounded metric spaces ............. 46 17.1 Precompact subsets of X .................... 48 Problems and Complements ....................... 49 II Measuring Sets 65 1 Partitioning open subsets of RN ................... 65 2 Limits of sets, characteristic functions, and σ -algebras ....... 67 3 Measures ............................... 68 3.1 Finite, σ -finite, and complete measures ............ 71 3.2 Some examples ......................... 71 4 Outer measures and sequential coverings .............. 72 4.1 The Lebesgue outer measure in ¡R·^ .............. 73 4.2 The Lebesgue-Stieltjes outer measure ............. 73 5 The Hausdorff outer measure in RN ................. 74 6 Constructing measures from outer measures ............. 76 7 The Lebesgue-Stieltjes measure on R ................ 79 7.1 Borei measures ......................... 80 8 The Hausdorff measure on RN .................... 80 9 Extending measures from semialgebras to σ -algebras ........ 82 9.1 On the Lebesgue-Stieltjes and Hausdorff measures ...... 84 10 Necessary and sufficient conditions for measurability ........ 84 11 More on extensions from semialgebras to σ -algebras ........ 86 12 The Lebesgue measure of sets in EN ................. 88 12.1 A necessary and sufficient condition of measurability ..... 88 13 A nonmeasurable set ......................... 90 Contents vii 14 Borei sets, measurable sets, and incomplete measures ........ 91 14.1 A continuous increasing function ƒ : [0, 1] —* [0,1]..... 91 14.2 On the preimage of a measurable set .............. 93 14.3 Proof of Propositions 14.1 and 14.2.............. 94 15 More on Borei measures ....................... 94 15.1 Some extensions to general Borei measures .......... 97 15.2 Regular Borei measures and Radon measures ......... 97 16 Regular outer measures and Radon measures ............ 98 16.1 More on Radon measures .................... 99 17 Vitali coverings ............................ 99 18 The Besicovitch covering theorem .................. 103 19 Proof of Proposition 18.2....................... 105 20 The Besicovitch measure-theoretical covering theorem .......107 Problems and Complements .......................110 III The Lebesgue Integral 123 1 Measurable functions ......................... 123 2 The Egorov theorem ......................... 126 2.1 The Egorov theorem in RN ................... 128 2.2 More on Egorov s theorem ................... 128 3 Approximating measurable functions by simple functions ...... 128 4 Convergence in measure ....................... 130 5 Quasi-continuous functions and Lusin s theorem .......... 133 6 Integral of simple functions ..................... 135 7 The Lebesgue integral of nonnegative functions ........... 136 8 Fatou s 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 12 Product of measures ......................... 142 13 On the structure of (Λ χ Β)..................... 144 14 The Fubini-Tonelli theorem ..................... 147 14.1 The Tonelli version of the Fubini theorem ........... 148 15 Some applications of the Fubini-Tonelli theorem .......... 148 15.1 Integrals in terms of distribution functions ........... 148 15.2 Convolution integrals ...................... 149 15.3 The Marcinkiewicz integral .................. 150 16 Signed measures and the Hahn decomposition ............ 151 17 The Radon-Nikodým theorem .................... 154 18 Decomposing measures ....................... 157 18.1 The Jordan decomposition ................... 157 18.2 The Lebesgue decomposition ................. 159 18.3 A general version of the Radon-Nikodým theorem ...... 160 Problems and Complements ....................... 160 viii Contents IV Topics on Measurable Functions of Real Variables 171 1 Functions of bounded variations ...................171 2 Dini derivatives ............................1^3 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 DM v .................186 9.1 Proof of Proposition 9.2....................188 10 Representing Dßv..........................189 10.1 Representing Dßv for v «/í ................. 189 10.2 Representing Dßv for v i. μ ..................191 11 The Lebesgue differentiation theorem ................191 11.1 Points of density ........................192 11.2 Lebesgue points of an integrable function ...........192 12 Regular families ...........................193 13 Convex functions ...........................194 14 Jensen s inequality ..........................196 15 Extending continuous functions ...................197 16 The Weierstrass approximation theorem ...............199 17 The Stone- Weierstrass theorem ...................200 18 Proof of the Stone-Weierstrass theorem ...............201 18.1 Proof of Stone s theorem ....................202 19 The Ascoli-Arzelà theorem . _...................203 19.1 Precompact subsets of C(E) ..................204 Problems and Complements .......................205 V The LP(E) Spaces 221 1 Functions in Ľ (£) and their norms .................221 1.1 The spaces Lp for 0 < p < 1 .................222 1.2 The spaces Ľ? for q < 0....................222 2 The Holder and Minkowski inequalities ...............223 3 The reverse Holder and Minkowski inequalities ...........224 4 More on the spaces Lp and their norms ...............225 4.1 Characterizing the norm ]\| ƒ \|\|p for 1 < p < oo ........225 4.2 The norm \|\| · Цоо for E of finite measure ............226 4.3 The continuous version of the Minkowski inequality .....227 5 Lp(£) for 1 < p < oo as normed spaces of equivalence classes . . 227 5.1 LP(E) for 1 < p < oo as a metric topological vector space . . 228 6 A metric topology for Lp( E) when 0 < p < 1 ...........229 6.1 Open convex subsets of LP(E) whenO < p < 1 .......229 7 Convergence in Lp(E) and completeness ..............230 8 Separating LP{E) by simple functions ................232 Contents ix 9 Weak convergence in L P(E) .....................234 9.1 A counterexample .......................234 10 Weak lower semicontinuity of the norm in LP(E) ..........235 11 Weak convergence and norm convergence ..............236 11.1 Proof of Proposition 11.1 for p > 2.............. 237 11.2 Proof of Proposition 11.1 for 1 < p < 2............ 237 12 Linear functionals in LP(E) ..................... 238 13 The Riesz representation theorem .................. 239 13.1 Proof of Theorem 13.1: The case where {Χ,Α, μ] is finite . .240 13.2 Proof of Theorem 13.1: The case where {Χ, Α, μ) is σ -nnite . 241 13.3 Proof of Theorem 13.1: The case where 1 < ρ < oo .....242 14 The Hanner and Clarkson inequalities ................243 14.1 Proof of Hanner s inequalities .................244 14.2 Proof of Clarkson s inequalities ................245 15 Uniform convexity of LP(E) for 1 < ρ < oo ............246 16 The Riesz representation theorem by uniform convexity ......247 16.1 Proof of Theorem 13.1: The case where 1 < ρ < oo ..... 247 16.2 The case where ρ = 1 and E is of finite measure ....... 248 16.3 The case where ρ = 1 and [Χ, Α, μ) is σ -finite........ 249 17 Bounded linear functional in LP(E) forO < ρ < 1......... 250 17.1 An alternate proof of Proposition 17.1............. 250 18 If E с KN and ρ є [l.oo), then LP(E) is separable ........ 251 18.1 L°°(£) is not separable .................... 254 19 Selecting weakly convergent subsequences ............. 254 20 Continuity of the translation in LP(E) for 1 < ρ < σο....... 255 21 Approximating functions in LP(E) with functions in C°°(E) . . . . 257 22 Characterizing precompact sets in LP(E) ..............260 Problems and Complements .......................262 VI Banach Spaces 275 1 Normed spaces ............................275 1.1 Seminorms and quotients ....................276 2 Finite- and infinite-dimensional normed spaces ...........277 2.1 A counterexample .......................277 2.2 The Riesz lemma ........................278 2.3 Finite-dimensional spaces ...................279 3 Linear maps and functionals .....................280 4 Examples of maps and functionals ..................282 4.1 Functionals ......._....................283 4.2 Linear functionals on C(E) ..................283 5 Kernels of maps and functionals ...................284 6 Equibounded families of linear maps ................285 6.1 Another proof of Proposition 6.1................286 7 Contraction mappings ........................286 7.1 Applications to some Fredholm integral equations .......287 χ Contents 8 The open mapping theorem .....................288 8.1 Some applications .......................289 8.2 The closed graph theorem ...................289 9 The Hahn-Banach theorem ......................290 10 Some consequences of the Hahn-Banach theorem .........292 10.1 Tangent planes .........................295 11 Separating convex subsets of X ...................295 12 Weak topologies ...........................297 12.1 Weakly and strongly closed convex sets ............299 13 Reflexive Banach spaces .......................300 14 Weak compactness ..........................301 14.1 Weak sequential compactness .................302 1 5 The weak* topology .........................303 16 The Alaoglu theorem .........................304 17 Hüben spaces .............................306 17.1 The Schwarz inequality .....................307 17.2 The parallelogram identity ...................307 18 Orthogonal sets, representations, and functionals ..........308 18.1 Bounded linear functionals on Я ................310 19 Orthonormal systems .........................310 19.1 The Bessel inequality ......................311 19.2 Separable Hubert spaces ....................312 20 Complete orthonormal systems ...................312 20.1 Equivalent notions of complete systems ............313 20.2 Maximal and complete orthonormal systems .........313 20.3 The Gram-Schmidt orthonormalization process ........314 20.4 On the dimension of a separable Hubert space .........314 Problems and Complements .......................314 VII Spaces of Continuous Functions, Distributions, and Weak Derivatives 325 1 Spaces of continuous functions ....................325 1.1 Partition of unity ........................326 2 Bounded linear functionals on CoiR^) ...............327 2.1 Remarks on functionals of the type (2.2) and (2.3)......327 2.2 Characterizing CO(RN)* ....................328 3 Positive linear functionals on Co (№.N) ................328 4 Proof of Theorem 3.3: Constructing the measure μ .........331 5 Proof of Theorem 3.3: Representing Γ as in (3.3)..........333 6 Characterizing bounded linear functionals on Co (RN) .......335 6.1 Locally bounded linear functionals on CoiR^) ........335 6.2 Bounded linear functionals on C<,(RN) ............336 7 A topology for C^ÍE) for an open set E ci ........... 337 8 A metric topology for C£°(£) ....................339 8.1 Equivalence of these topologies ................340 Contents xi 8.2 Đ(£) is not complete ......................341 9 A topology for C^i/O for a compact set К с £ .......... 341 9.1 A metric topology for C~( К ).................342 9.2 V(K) is complete ........................342 10 Relating the topology of D (£) to the topology of V(K) ......343 10.1 Noncompleteness of D(E) ...................344 11 The Schwartz topology of V (£)...................344 12 £>(£) is complete ...........................346 12.1 Cauchy sequences in V(E) ...................347 12.2 The topology of V(E) is not metrizable ............347 13 Continuous maps and functionals ..................348 13.1 Distributionson E....................... 348 13.2 Continuous linear maps T : £>(£) -* ЩЕ).......... 349 14 Distributional derivatives .......................349 14.1 Derivatives of distributions ...................350 14.2 Some examples .........................350 14.3 Miscellaneous remarks .....................351 15 Fundamental Solutions ........................352 15.1 The fundamental solution of the wave operator ........352 15.2 The fundamental solution of the Laplace operator .......354 16 Weak derivatives and main properties ................355 17 Domains and their boundaries ....................358 17.1 Э£ of classC1 .........................358 17.2 Positive geometric density ...................358 17.3 The segment property ......................358 17.4 The cone property .......................359 17.5 On the various properties of ЭЕ ................359 18 More on smooth approximations ...................359 19 Extensions into RN ..........................361 20 The chain rule ............................363 21 Steklov averagings ..........................365 22 Characterizing Wlp(E) for 1 < ρ < oo ..............367 22.1 Remarkson Wloc(E) .....................368 23 The Rademacher theorem ......................368 Problems and Complements .......................371 VIII Topics on Integrable Functions of Real Variables 375 1 Vitali-type coverings .........................375 2 The maximal function ........................377 3 Strong Lp estimates for the maximal function ............379 3.1 Estimates of weak and strong type ...............380 4 The Calderón-Zygmund decomposition theorem ..........381 5 Functions of bounded mean oscillation ...............383 6 Proof of Theorem 5.1.........................384 7 The sharp maximal function .....................387 xii Contents 8 Proof of the Fefferman-Stein theorem ................388 9 The Marcinkiewicz interpolation theorem ..............390 9.1 Quasi-linear 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 14.1 Approximations by simple functions ..............400 15 Reduction to a finite union of intervals ................402 16 Proof of Theorem 14.1: The case where Τ + S < R......... 404 17 Proof of Theorem 14.1: The case where S + T > R ........ 404 17.1 Proof of Lemma 17.1......................407 18 Hardy s inequality ..........................407 19 A convolution-type inequality ....................409 19.1 Some reductions ........................409 20 Proof of Theorem 19.1........................410 21 An equivalent form of Theorem 19.1.................411 22 An N -dimensional version of Theorem 21.1.............412 23 Lp estimates of Riesz potentials ...................413 24 The limiting case p = N.......................415 Problems and Complements .......................417 IX Embeddings of Wl P(E) into Ľ>(E) 423 1 Multiplicative embeddings of W,] P(E) ...............423 2 Proof of Theorem 1.1 for N = 1...................425 3 Proof of Theorem 1.1 for 1 < p < N................425 4 Proof of Theorem 1.1 for 1 < p < N, concluded ..........428 5 Proof of Theorem 1.1 for p > N > 1 ................428 5.1 Estimate of /ι (л. R)...................... 429 5.2 Estimate of/2 (.x, R)...................... 430 6 Proof of Theorem 1.1 for p > N > 1, concluded ..........430 7 On the limiting case p = N .....................431 8 Embeddings of W p (E)....................... 432 9 Proof of Theorem 8.1.........................433 10 Poincaré inequalities .........................435 10.1 The Poincaré inequality ....................435 10.2 Multiplicative Poincaré inequalities ..............437 1 1 The discrete isoperimetric inequality .................438 12 Morrey spaces ............................439 12.1 Embeddings for functions in the Morrey spaces ........440 13 Limiting embedding of W ΙΝ(£) .................. 441 14 Compact embeddings .........................443 15 Fractional Sobolev spaces in RN ...................445 16 Traces .................................447 Contents xiii 17 Traces and fractional Sobolev spaces ................448 18 Traces on 3E of functions in W]P(E) ................450 18.1 Traces and fractional Sobolev spaces .............453 19 Multiplicative embeddings of W^P(E) ...............453 20 Proof of Theorem 19.1: A special case ................456 21 Constructing a map between E and Q : Part 1............458 22 Constructing a map between E and Q: Part 2............460 23 Proof of Theorem 19.1, concluded ..................463 Problems and Complements .......................464 References 469 Index 473
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title	Real analysis
title_auth	Real analysis
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title_full	Real analysis Emmanuele DiBenedetto
title_fullStr	Real analysis Emmanuele DiBenedetto
title_full_unstemmed	Real analysis Emmanuele DiBenedetto
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work_keys_str_mv	AT dibenedettoemmanuele realanalysis

Verfügbarkeit

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Per Ortsleihe bestellen Per Ortsleihe bestellen Inhaltsverzeichnis

An UBR ausleihen
Ausleihen an der UBR sind mit eigenem Ausweis möglich.