分析 第三卷（英文版） 出版时间：2012年版 内容简介 This third volume concludes our introduction to analysis， where in we finish laying the groundwork needed for further study of the subject. As with the first two， this volume contains more material than can treated in a single course. It is therefore important in preparing lectures to choose a suitable subset of its content; the remainder can be treated in seminars or left to independent study. For a quick overview of this content， consult the table of contents and the chapter introductions. 目录 Foreword Chapter Ⅸ Elements of measure theory 1 Measurable spaces σ-algebras The Borel σ-algebra The second countability axiom Generating the Borel a-algebra with intervals Bases of topological spaces The product topology Product Borel a-algebras Measurability of sections 2 Measures Set functions Measure spaces Properties of measures Null sets Outer measures The construction of outer measures The Lebesgue outer measure The Lebesgue-Stieltjes outer measure Hausdorff outer measures 4 Measurable sets Motivation The a-algebra of/μ*-measurable sets Lebesgue measure and Hausdorff measure Metric measures 5 The Lebasgue measure The Lebesgue measure space The Lebesgue measure is regular A characterization of Lebesgue measurability Images of Lebesgue measurable sets The Lebesgue measure is translation invariant A characterization of Lebesgue measure The Lebesgue measure is invariant under rigid motions The substitution rule for linear maps Sets without Lebesgue measure Chapter Ⅹ Integration theory 1 Measurable functions Simple functions and measurable functions A measurability criterion Measurable R-valued functions The lattice of measurable T-valued functions Pointwise limits of mensurable functions Radon measures 2 Integrable fuuctions The integral of a simple function The L1-seminorm The Bochner-Lebesgue integral The completeness of L1 Elementary properties of integrals Convergence in L1 3 Convergence theorems Integration of nonnegative T-valued functions The monotone convergence theorem Fatou''''s lemma Integration of R-valued functions Lebesgue''''s dominated convergence theorem Parametrized integrals 4 Lebesgue spaces Essentially bounded functions The Holder and Minkowski inequalities Lebesgue spaces are complete Lp-spaces Continuous functions with compact support Embeddings Continuous linear functionals on Lp 5 The n-dimensional Bochner-Lebesgue integral Lebesgue measure spaces The Lebesgue integral of absolutely integrable functions A characterization of Riemann integrable functions 6 Fubiul''''s theorem Maps defined almost everywhere Cavalieri''''s principle Applications of Cavalieri''''s principle Tonelli''''s theorem Fubini''''s theorem for scalar functions Fubini''''s theorem for vector-vained functions Minkowski''''s inequality for integrals A characterization of Lp （Rm+n, E） A trace theorem 7 The convolution Defining the convolution The translation group Elementary properties of the convolution Approximations to the identity Test functions Smooth partitions of unity Convolutions of E-valued functions Distributions Linear differential operators Weak derivatives 8 The substitution rule Pulling back the Lebesgue measure The substitution rule: general case Plane polar coordinates Polar coordinates in higher dimensions Integration of rotationally symmetric functions The substitution rule for vector-valued functions 9 The Fourier transform Definition and elementary properties The space of rapidly decreasing functions The convolution algebra S Calculations with the Fourier transform The Fourier integral theorem Convolutions and the Fourier transform Fourier multiplication operators Plancherel''''s theorem Symmetric operators The Heisenberg uncertainty relation Chapter Ⅺ Manifolds and differential forms 1 Submanifolds Definitions and elementary properties Submersions Submanifo]ds with boundary Local charts Tangents and normals The regular value theorem One-dimensional manifolds Partitions of unity 2 MultUinear algebra Exterior products Pull backs The volume element The Riesz isomorphism The Hodge star operator Indefinite inner products Tensors 3 The local theory of differential forms Definitions and basis representations Pull backs The exterior derivative The Poincare lemma Tensors 4 Vector fields and differential forms Vector fields Local basis representation Differential forms Local representations Coordinate transformations The exterior derivative Closed and exact forms Contractions Orientability Tensor fields 5 Riemannian metrics The volume element Riemannian manifolds The Hodge star The codifferential 6 Vector analysis The Riesz isomorphism The gradient The divergence The Laplace-Beltrami operator The curl The Lie derivative The Hodge-Laplace operator The vector product and the curl Chapter Ⅻ Integration on manifolds 1 Volume measure The Lebesgue a-algebra of M The defiaition of the volume measure Properties Integrability Calculation of several volumes 2 Integration of differential forms Integrals of m-forms Restrictions to submanifolds The transformation theorem Fubini''''s theorem Calculations of several integrals Flows of vector fields The transport theorem 3 Stokes''''s theorem Stokes''''s theorem for smooth manifolds Manifolds with singularities Stokes''''s theorem with singularities Planar domains Higher-dimensional problems Homotopy invariance and applications Gauss''''s law Green''''s formula The classical Stokes''''s theorem The star operator and the coderivative References
