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
