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实分析（影印版）

暂无简介
作者：
Emmanuele DiBenedetto
定价：
60.00元
出版时间：
2007-10-30
ISBN：
978-7-04-022665-2
物料号：
22665-00
读者对象：
高等教育
一级分类：
数学与统计学类
二级分类：
数学与应用数学专业课
三级分类：
其他课程
重点项目：
暂无
版面字数：
560.000千字
开本：
16开
全书页数：
485页
装帧形式：
平装
  • Preface
  • Acknowledgments
  • Preliminaries
    • 1 Countable sets
    • 2 The Cantor set
    • 3 Cardinality
    • 3.1 Some examples
    • 4 Cardinality of some infinite Cartesian products
    • 5 Ordenngs, the maximal principle, and the axiom of choice
    • 6 Well-ordering
    • 6.1 The first uncountable
    • Problems and Complements
  • I Topologies and Metric Spaces
    • 1 Topological spaces
      • 1.1 Hausdorff and normal spaces
    • 2 Urysohn's lemma
    • 3 The Tietze extension theorem
    • 4 Bases,axioms of countability,and product topologies
      • 4.1 Product topologies
    • 5 Compact topological spaces
      • 5.1 Sequentially compact topological spaces
    • 6 Compact subsets of RN
    • 7 Continuous functions on countably compact spaces
    • 8 Products of compact spaces
    • 9 Vector spaces
      • 9.1 Convex sets
      • 9.2 Linear maps and isomorphisms
    • 10 Topological vector spaces
      • 10.1 Boundedness and continuity
    • 11 Linear functionals
    • 12 Finite-dimensional topological vector spaces
      • 12.1 Locally compact spaces
    • 13 Metric spaces
      • 13.1 Separation and axioms of countability
      • 13.2 Equivalent metrics
      • 13.3 Pseudometrics
    • 14 Metric vector spaces
      • 14.1 Maps between metric spaces
    • 15 Spaces of continuous functions
      • 15.1 Spaces of continuously differentiable functions
    • 16 On the structure of a complete metric space
    • 17 Compact and totally bounded metric spaces
      • 17.1 Precompact subsets of X
  • Problems and Complements
  • Ⅱ Measuring Sets
    • 1 Partitioning open subsets of RN
    • 2 Limits of sets, characteristic functions, and ●-algebras
    • 3 Measures
      • 3.1 Finite, ●-finite, and complete measures
      • 3.2 Some examples
    • 4 Outer measures sequential coverings
      • 4.1 The Lebes and outer measure in RN
      • 4 2 The Lebesgue-Stieltjes outer measure
    • 5 The Hausdorff outer measure in RN
    • 6 Constructing measures from outer measures
    • 7 The Lebesgue-Stieltjes measure on R 7.1 Borel measures
    • 8 The Hausdorff measure on RN
    • 9 Extending measures from semialgebras to a-algebras
      • 9.1 On the Lebesgue-Stieltjes and Hausdorff measures
    • 10 Necessary and sufficient conditions for measurability
    • 11 More on extensions from semialgebras to a -algebras
    • 12 The Lebesgue measure of sets in Rn
      • 12.1 A necessary and sufficient condition of measurability
    • 13 A nonmeasurable set
    • 14 Borel sets, measurable sets, and incomplete measures
      • 14.1 A continuous increasing function f：[0,1]●[0,1]
      • 14.2 On the preimage of a measurable set
      • 14.3 Proof of Propositions 14.1 and 14.2
    • 15 More on Borel measures
      • 15.1 Some extensions to general Borel measures
      • 15.2 Regular Bore] measures and Radon measures
    • 16 Regular outer measures and Radon measures
      • 16.1 More on Radon measures
    • 17 Vitali coverings
    • 18 The Besicovitch covering theorem
    • 19 Proof of Proposition 18.2
    • 20 The Besicovitch measure-theoretical covering theorem
  • Problems and Complements
  • III The Lebesgue Integral
    • 1 Measurable functions
    • 2 The Egorov theorem
      • 2.1 The Egorov theorem in RN
      • 2.2 More on Egorov's theorem
    • 3 Approximating measurable functions by simple functions
    • 4 Convergence in measure
    • 5 Quasi-continuous functions and Lusin's theorem
    • 6 Integral of simple functions
    • 7 The Lebesgue integral of nonnegative functions
    • 8 Fatou's lemma and the monotone convergence theorem
    • 9 Basic properties of the Lebesgue integral
    • 10 Convergence theorems
    • 11 Absolute continuity of the integral
    • 12 Product of measures
    • 13 On the structure of(A×B)
    • 14 The Fubini-Tonelli theorem
      • 14.1 The Tonelli version of the Fubini theorem
    • 15 Some applications of the Fubini-Tonelli theorem
      • 15.1 Integrals in terms of distribution functions
      • 15.2 Convolution integrals
      • 15.3 The Marcinkiewicz integral
    • 16 Signed measures and the Hahn decomposition．
    • 17 The Radon-Nikod●m theorem
    • 18 Decomposing measures
      • 18.1 The Jordan decomposition
      • 18.2 The Lebesgue decomposition
      • 18.3 A general version of the Radon-Nikod●m theorem
    • Problems and Complements
  • Ⅳ Topics on Measurable Functions of Real Variables
    • 1 Functions of bounded variations
    • 2 Dini derivatives
    • 3 Differentiating functions of bounded variation
    • 4 Differentiating series of monotone functions
    • 5 Absolutely continuous functions
    • 6 Density of a measurable set
    • 7 Derivatives of integrals
    • 8 Differentiating Radon measures
    • 9 Existence and measurability of D●●
      • 9.1 Proof of Proposition 9.2
    • 10 Representing D●●
      • 10.1 Representing ●●for v●u
      • 10.2 Representing D●●for●●●
    • 11 The Lebesgue differentiation theorem
      • 11.1 Points of density
      • 11.2 Lebesgue points of an integrable function
    • 12 Regular families
    • 13 Convex functions
    • 14 Jensen's inequality
    • 15 Extending continuous functions
    • 16 The Weierstrass approximation theorem
    • 17 The Stone-Weierstrass theorem
    • 18 Proof of the Stone-Weierstrass theorem
      • 18.1 Proof of Stone's theorem
    • 19 The Ascoli-Arzelàtheorem
      • 19.1 Precompact subsets of C(●）
  • Problems and Complements
  • V The LP(E) Spaces
    • 1 Functions in LP(E) and their norms
      • 1.1 The spaces LP for 0＜P＜1
      • 1.2 The spaces L9 for q＜0
    • 2 The Wider and Minkowski inequalities
    • 3 The reverse H61der and Minkowski inequalities
    • 4 More on the spaces LP and their norms
      • 4.1 Characterizing the norm●●●for 1●p＜●
      • 4.2 The norm●●●●for E of finite measure
      • 4.3 The continuous version of the Minkowski inequality
    • 5 LP(E) for1●p < oo as normed spaces of equivalence classes
      • 5.1 LP(E) for 1●●●as a matrir tnpological vector space a metric topological vector siDace
    • 6 A metric topology for LP(E) when 0＜P＜1
      • 6.1 Open convex subsets of LP(E) when 0＜p＜1
    • 7 Convergence in LP(E) and completeness
    • 8 Separating LP(E) by simple functions
    • 9 Weak convergence in LP(E)
      • 9.1 A counterexample
    • 10 Weak lower semicontinuity of the norm in LP(E)
    • 11 Weak convergence and norm convergence
      • 11.1 Proof of Proposition 11.1 for p●2
      • 11.2 Proof of Proposition 11.1 for 1＜P＜2
    • 12 Linear functionals in LP(E)
    • 13 The Riesz representation theorem
      • 13.1 Proof of Theorem 13.1：The case where{X,A,u}is finite
      • 13.2 Proof of Theorem 13.1：The case where{X,A，u} is σ-finite
      • 13.3 Proof of Theorem13.1：The case where 1＜P＜●
    • 14 The Hanner and Clarkson inequalities
      • 14.1 Proof of Hanner's inequalities
      • 14.2 Proof of Clarkson's inequalities
    • 15 Uniform convexity of LP(E) for 1＜P＜●
    • 16 The Riesz representation theorem by uniform convexity
      • 16.1 Proof of Theorem 13.1：The case where 1＜P＜●
      • 16.2 The case where p＝1 and E is of finite measure
      • 16.3 The case where p＝1 and{X,A,u}is σ-finite
    • 17 Bounded linear functional in LP (E) for 0＜P＜1
      • 17.1 An alternate proof of Proposition 17.1
    • 18 If E●RN and p●[1，●),then LP(E) is separable
      • 18.1 ●●●is not separable
    • 19 Selecting weakly convergent subsequences
    • 20 Continuity of the translation in LP(E) for 1 < p＜00
    • 21 Approximating functions in LP(E) with functions in C●(E)
    • 22 Characterizing precompact sets in LP(E)
  • Problems and Complements
  • VI Banach Spaces
    • 1 Normed spaces
      • 1.1 Seminorms and quotients
    • 2 Finite- and infinite-dimensional normed spaces
      • 2.1 A counterexample
      • 2.2 The Riesz lemma
      • 2.3 Finite-dimensional spaces
    • 3 Linear maps and functionals
    • 4 Examples of maps and functionals
      • 4.1 Functionals
      • 4.2 Linear functionals on C(●)
    • 5 Kernels of maps and functionals
    • 6 Equibounded families of linear maps
      • 6.1 Another proof of Proposition 6.1
    • 7 Contraction mappings
      • 7.1 Applications to some Fredholm integral equations
    • 8 The open mapping theorem
      • 8.1 Some applications
      • 8.2 The closed graph theorem
    • 9 The Hahn-Banach theorem
    • 10 Some consequences of the Hahn-Banach theorem
      • 10.1 Tangent planes
    • 11 Separating convex subsets of X
    • 12 Weak topologies
      • 12.1 Weakly and strongly closed convex sets
    • 13 Reflexive Banach spaces
    • 14 Weak compactness
      • 14.1 Weak sequential compactness
    • 15 The weak' topology
    • 16 The Alaoglu theorem
    • 17 Hilbert spaces
      • 17.1 The Schwarz inequality
      • 17.2 The parallelogram identity
    • 18 Orthogonal sets,representations,and functionals
      • 18.1 Bounded linear functionals on H
    • 19 Orthonormal systems
      • 19.1 The Bessel inequality
      • 19.2 Separable Hilbert spaces
    • 20 Complete orthonormal systems
      • 20.1 Equivalent notions of complete systems
      • 20.2 Maximal and complete orthonormal systems
      • 20.3 The Gram-Schmidt orthonormalization process
      • 20.4 On the dimension of a separable Hilbert space Problems and Complements
  • VII Spaces of Continuous Functions, Distributions, and Weak Derivatives
    • 1 Spaces of continuous functions
      • 1.1 Partition of unity
    • 2 Bounded linear functionals on Ca(RN)
      • 2.1 Remarks on functionals of the type (2.2) and (2.3)
      • 2.2 Characterizing Co(RN)'
    • 3 Positive linear functionals on Co(RN)
    • 4 Proof of Theorem 3.3:Constructing the measure u
    • 5 Proof of Theorem 3.3:Representing T as in (3.3)
    • 6 Characterizing bounded linear functionals on Co(RN)
      • 6.1 Locally bounded linear functionals on Co(RN)
      • 6.2 Bounded linear functionals on Co(RN)
    • 7 A topology for ●(E) for an open set E●RN
    • 8 A metric topology for ●(E)
      • 8.1 Equivalence of these topologies
      • 8.2 D(E) is not complete
    • 9 A topology for ●●(K) for a compact set ●●●
      • 9.1 A metric topology for ●(K)
      • 9.2 D（k） is complete
    • 10 Relating the topology of D(E) to the topology of D(K)
      • 10.1 Noncompleteness of D(E)
    • 11 The Schwartz topology of D(E)
    • 12 D(E) is complete
      • 12.1 Cauchy sequences in D（E）
      • 12.2 The topology of D(E) is not metrizable
    • 13 Continuous maps and functionals
      • 13.1 Distributions on E
      • 13.2 Continuous linear maps T：D(E)●D(E)
    • 14 Distributional derivatives
      • 14.1 Derivatives of distributions
      • 14.2 Some examples
      • 14.3 Miscellaneous remarks
    • 15 Fundamental Solutions
      • 15.1 The fundamental solution of the wave operator
      • 15.2 The fundamental solution of the Laplace operator
    • 16 Weak derivatives and main properties
    • 17 Domains and their boundaries
      • 17.1 ●E of class C1
      • 17.2 Positive geometric density
      • 17.3 The segment property
      • 17.4 The cone property
      • 17.5 On the various properties of ●E
    • 18 More on smooth approximations
    • 19 Extensions into RN
    • 20 The chain rule
    • 21 Steklov averagings
    • 22 Characterizing ●(E) for 1＜P＜●
      • 22.1 Remarks on ●(E)
    • 23 The Rademacher theorem
  • Problems and Complements
  • VIII Topics on Integrable Functions of Real Variables
    • 1 Vitali-type coverings
    • 2 The maximal function
    • 3 Strong LP estimates for the maximal function
      • 3.1 Estimates of weak and strong type
    • 4 The Calderbn-Zygmund decomposition theorem
    • 5 Functions of bounded mean oscillation
    • 6 Proof of Theorem 5.1
    • 7 The sharp maximal function
    • 8 Proof of the Fefferman-Stein theorem
    • 9 The Marcinkiewicz interpolation theorem
      • 9.1 Quasi-linear maps and interpolation
    • 10 Proof of the Marcinkiewicz theorem
    • 11 Rearranging the values of a function
    • 12 Basic properties of rearrangements
    • 13 Symmetric rearrangements
    • 14 A convolution inequality for rearrangements
      • 14.1 Approximations by simple functions
    • 15 Reduction to a finite union of intervals
    • 16 Proof of Theorem 14.1：The case where T＋S<R
    • 17 Proof of Theorem 14.1：The case where S＋T＞R
      • 17.1 Proof of Lemma 17.1
    • 18 Hardy's inequality
    • 19 A convolution-type inequality
      • 19.1 Some reductions
    • 20 Proof of Theorem 19.1
    • 21 An equivalent form of Theorem 19.1
    • 22 An N-dimensional version of Theorem 21.1
    • 23 LP estimates of Riesz potentials
    • 24 The limiting case p＝N
  • Problems and Complements
  • Ⅸ Embeddings of W1,P(E) into Lq(E)
    • 1 Multiplicative embeddings of ●●（E）
    • 2 Proof of Theorem 1.1 for N＝1
    • 3 Proof of Theorem 1.1 for 1●p＜N
    • 4 Proof of Theorem 1.l for 1●p＜N，concluded
    • 5 Proof of Theorem 1.1 forp●N＞1
      • 5.1 Estimate of/I1(x，R)
      • 5.2 Estimate of I2(x，R)
    • 6 Proof of Theorem 1.1 for p●N＞1, concluded
    • 7 On the limiting case p＝N
    • 8 Emheddings of ●（E）
    • 9 Proof of Theorem 8.1
    • 10 Poincard inequalities
      • 10.1 The Poincard inequality
      • 10.2 Multiplicative Poincare inequalities
    • 11 The discrete isoperimetric inequality
    • 12 Morrey spaces
      • 12.1 Embeddings for functions in the Morrev spaces
    • 13 Limiting embedding of ●(E)
    • 14 Compact embeddings
    • 15 Fractional Sobolev spaces in RN
    • 16 Traces
    • 17 Traces and fractional Sobolev spaces
    • 18 Traces on aE of functions in W●(E)
      • 18.1 Traces and fractional Sobolev spaces
    • 19 Multiplicative embeddings
    • 20 Proof of Theorem 19.1：A special case
    • 21 Constructing a map between E and Q: Part 1
    • 22 Constructing a map between E and Q: Part 2
    • 23 Proof of Theorem 19.1，concluded
  • Problems and Complements
  • References
  • Index

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