包邮实分析

foreword
introduction
1 fourier series: completion
2 limits of continuous functions
3 length of curves
4 differentiation and integration
5 the problem of measure
chapter 1. measure theory
1 preliminaries
2 the exterior measure
3 measurable sets and the lebesgue measure
4 measurable functions
4.1 definition and basic properties
4.2 approximation by simple functions or step functions
4.3 littlewood's three principles
5* the brunn-minkowski inequality
6 exercises
7 problems
chapter 2. integration theory
1 the lebesgue integral: basic properties and convergence theorems
2 the space l1 of integrable functions
3 fubini's theorem
3.1 statement and proof of the theorem
3.2 applications of fubini's theorem
4* a fourier inversion formula
5 exercises
6 problems
chapter 3. differentiation and integration
1 differentiation of the integral
1.1 the hardy-littlewood maximal function
1.2 the lebesgue differentiation theorem
2 good kernels and approximations to the identity
3 differentiability of functions
3.1 functions of bounded variation
3.2 absolutely continuous functions
3.3 differentiability of jump functions
4 rectifiable curves and the isoperimetric inequality
4.1 minkowski content of a curve
4.2* isoperimetrie inequality
5 exercises
6 problems
chapter 4. hilbert spaces: an introduction
1 the hilbert space l2
2 hilbert spaces
2.1 orthogonality
2.2 unitary mappings
2.3 pre-hilbert spaces
3 fourier series and fatou's theorem
3.1 fatou's theorem
4 closed subspaees and orthogonal projections
5 linear transformations
5.1 linear flmetionals and the riesz representation the-orem
5.2 adjoints
5.3 examples
6 compact operators
7 exercises
8 problems
chapter 5. hilbert spaces: several examples
1 the fourier transform on l2
2 the hardy space of the upper half-plane
3 constant coefficient partial differential equations
3.1 weak solutions
3.2 the main theorem and key estimate
4* the dirichlet principle
4.1 harmonic functions
4.2 the boundary value problem and diriehlet's principle
5 exercises
6 problems
chapter 6.abstract measure and integration theory
chapter 7.hausdorff measure and fractals
notes and references
bibliography
symbol glossary
index