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简介
本书是分析领域内的一部经典著作。毫不夸张地说，掌握了本书，对数学的理解将会上一个新台阶。全书体例优美，实用性很强，列举的实例简明精彩。无论实分析部分还是复分析部分，基本上对所有给出的命题都进行了论证。另外，书中还附有大量设计巧妙的习题——这些习题可以真实地检测出读者对课程的理解程度，有的还要求对正文中的原理进行论证。在第3版中，作者对一些新的课题进行了讨论，并力求全书条理清晰。本书适合作为高等院校数学专业研究生教材。
目录
preface
prologue: the exponential function
chapter 1 abstract integration
set-theoretic notations and terminology
the concept of measurability
simple functions
elementary properties of measures
arithmetic in [0, ]
integration of positive functions
integration of complex functions
the role played by sets of measure zero
exercises
chapter 2 positive borel measures
vector spaces
topological preliminaries
the riesz representation theorem
regularity properties of borel measures
lebesgue measure
continuity properties of measurable functions
exercises
.chapter 3 lp-spaces
convex functions and inequalities
the lp-spaces
approximation by continuous functions
exercises
chapter 4 elementary hilbert space theory
inner products and linear functionals
orthonormal sets
trigonometric series
exercises
chapter 5 examples of banach space techniques
banach spaces
consequences of baire's theorem
fourier series of continuous functions
fourier coefficients of l1-functions
the hahn-banach theorem
an abstract approach to the poisson integral
exercises
chapter 6 complex measures
total variation
absolute continuity
consequences of the radon-nikodym theorem
bounded linear functionals on lp
the riesz representation theorem
exercises
chapter 7 differentiation
derivatives of measures
the fundamental theorem of calculus
differentiable transformations
exercises
chapter 8 integration on product spaces
measurability on cartesian products
product measures
the fubini theorem
completion of product measures
convolutions
distribution functions
exercises
chapter 9 fourier transforms
formal properties
the inversion theorem
the plancherel theorem
the banach algebra l1
exercises
chapter 10 elementary properties of holomorphic
functions
complex differentiation
integration over paths
the local cauchy theorem
the power series representation
the open mapping theorem
the global cauchy theorem
the calculus of residues
exercises
chapter 11 harmonic functions
the cauchy-riemann equations
the poisson integral
the mean value property
boundary behavior of poisson integrals
representation theorems
exercises
chapter 12 the maximum modulus principle
introduction
the schwarz lemma
the phragmen-lindelof method
an interpolation theorem
a converse of the maximum modulus theorem
exercises
chapter 13 approximation by rational functions
preparation
runge's theorem
the mittag-leffler theorem
simply connected regions
exercises
chapter 14 conformal mapping
preservation of angles
linear fractional transformations
normal families
the riemann mapping theorem
the class
continuity at the boundary
conformai mapping of an annulus
exercises
chapter 15 zeros of holomorphic functions
infinite products
the weierstrass factorization theorem
an interpolation problem
jensen's formula
blaschke products
the miintz-szasz theorem
exercises
chapter 16 analytic continuation
regular points and singular points
continuation along curves
the monodromy theorem
construction of a modular function
the picard theorem
exercises
chapter 17 hp-spaces
subharmonic functions
the spaces hr and n
the theorem of f. and m. riesz
factorization theorems
the shift operator
conjugate functions
exercises
chapter 18 elementary theory of banach algebras
introduction
the invertible elements
ideals and homomorphisms
applications
exercises
chapter 19 holomorphic fourier transforms
introduction
two theorems of paley and wiener
quasi-analytic classes
the denjoy-carleman theorem
exercises
chapter 20 uniform approximation by polynomials
introduction
some iemmas
mergelyan's theorem
exercises
appendix: hausdorff's maximality theorem
notes and comments
bibliography
list of special symbols
index
prologue: the exponential function
chapter 1 abstract integration
set-theoretic notations and terminology
the concept of measurability
simple functions
elementary properties of measures
arithmetic in [0, ]
integration of positive functions
integration of complex functions
the role played by sets of measure zero
exercises
chapter 2 positive borel measures
vector spaces
topological preliminaries
the riesz representation theorem
regularity properties of borel measures
lebesgue measure
continuity properties of measurable functions
exercises
.chapter 3 lp-spaces
convex functions and inequalities
the lp-spaces
approximation by continuous functions
exercises
chapter 4 elementary hilbert space theory
inner products and linear functionals
orthonormal sets
trigonometric series
exercises
chapter 5 examples of banach space techniques
banach spaces
consequences of baire's theorem
fourier series of continuous functions
fourier coefficients of l1-functions
the hahn-banach theorem
an abstract approach to the poisson integral
exercises
chapter 6 complex measures
total variation
absolute continuity
consequences of the radon-nikodym theorem
bounded linear functionals on lp
the riesz representation theorem
exercises
chapter 7 differentiation
derivatives of measures
the fundamental theorem of calculus
differentiable transformations
exercises
chapter 8 integration on product spaces
measurability on cartesian products
product measures
the fubini theorem
completion of product measures
convolutions
distribution functions
exercises
chapter 9 fourier transforms
formal properties
the inversion theorem
the plancherel theorem
the banach algebra l1
exercises
chapter 10 elementary properties of holomorphic
functions
complex differentiation
integration over paths
the local cauchy theorem
the power series representation
the open mapping theorem
the global cauchy theorem
the calculus of residues
exercises
chapter 11 harmonic functions
the cauchy-riemann equations
the poisson integral
the mean value property
boundary behavior of poisson integrals
representation theorems
exercises
chapter 12 the maximum modulus principle
introduction
the schwarz lemma
the phragmen-lindelof method
an interpolation theorem
a converse of the maximum modulus theorem
exercises
chapter 13 approximation by rational functions
preparation
runge's theorem
the mittag-leffler theorem
simply connected regions
exercises
chapter 14 conformal mapping
preservation of angles
linear fractional transformations
normal families
the riemann mapping theorem
the class
continuity at the boundary
conformai mapping of an annulus
exercises
chapter 15 zeros of holomorphic functions
infinite products
the weierstrass factorization theorem
an interpolation problem
jensen's formula
blaschke products
the miintz-szasz theorem
exercises
chapter 16 analytic continuation
regular points and singular points
continuation along curves
the monodromy theorem
construction of a modular function
the picard theorem
exercises
chapter 17 hp-spaces
subharmonic functions
the spaces hr and n
the theorem of f. and m. riesz
factorization theorems
the shift operator
conjugate functions
exercises
chapter 18 elementary theory of banach algebras
introduction
the invertible elements
ideals and homomorphisms
applications
exercises
chapter 19 holomorphic fourier transforms
introduction
two theorems of paley and wiener
quasi-analytic classes
the denjoy-carleman theorem
exercises
chapter 20 uniform approximation by polynomials
introduction
some iemmas
mergelyan's theorem
exercises
appendix: hausdorff's maximality theorem
notes and comments
bibliography
list of special symbols
index
Real and complex analysis = 实分析与复分析 / 3rd ed.
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