实分析（影印版）

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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