Analysis III

This third volume concludes our introduction to analysis, wherein we ?nish laying the groundwork needed for further study of the subject. As with the ?rst 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. Thisbookisalsosuitableasbackgroundforothercoursesorforselfstudy. We hope that its numerous glimpses into more advanced analysis will arouse curiosity and so invite students to further explore the beauty and scope of this branch of mathematics. In writing this volume, we counted on the invaluable help of friends, c- leagues, sta?, and students. Special thanks go to Georg Prokert, Pavol Quittner, Olivier Steiger, and Christoph Walker, who worked through the entire text cr- ically and so helped us remove errors and make substantial improvements. Our thanks also goes out to Carlheinz Kneisel and Bea Wollenmann, who likewise read the majority of the manuscript and pointed out various inconsistencies. Without the inestimable e?ortofour “typesetting perfectionist”, this volume could not have reached its present form: her tirelessness and patience with T X E and other software brought not only the end product, but also numerous previous versions,to a high degree of perfection. For this contribution, she has our greatest thanks. Foreword Contents Chapter IX Elements of measure theory 1 Measurable spaces σ-algebras The Borel σ-algebra The second countability axiom Generating the Borel σ-algebra with intervals Bases of topological spaces The product topology Product Borel σ-algebras Measurability of sections 2 Measures Set functions Measure spaces Properties of measures Null sets 3 Outer measures The construction of outer measures The Lebesgue outer measure The Lebesgue–Stieltjes outer measure Hausdorff outer measures 4 Measurable sets Motivation The σ-algebra of μ∗-measurable sets Lebesgue measure and Hausdorff measure Metric measures 5 The Lebesgue 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 R-valued functions Pointwise limits of measurable functions Radon measures 2 Integrable functions 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 R-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 Hölder 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 Fubini’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-valued 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 Submanifolds with boundary Local charts Tangents and normals The regular value theorem One-dimensional manifolds Partitions of unity 2 Multilinear 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 Poincaré 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 σ-algebra of M The definition 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 Index This book is the first of a three volume introduction to analysis. It is distinguished by its modern and clear presentation, concentrating always on the essential concepts. In contrast to most other textbooks, there is no artificial separation between the theories of one variable and that of many variables. Emphasis is placed on the early development of a solid foundation in topology. As well, the basics of complex analysis are covered. This book is directed primarily to the students and instructors of beginning courses in analysis. But, with the many examples, exercises and the supplementary material, it is also suitable for self-study, as preparation for advanced study, and as a basis for other research in mathematics and physics."This textbook provides an outstanding introduction to analysis. It is distinguished by its high level of presentation and its focus on the essential.''Zeitschrift für Analysis und ihre Anwendung 18, No. 4 (G. Berger, review of the first German edition)"One advantage of this presentation is that the power of the abstract concepts are convincingly demonstrated using concrete applications.''W. Grölz, review of the first German edition The third and last volume of this work is devoted to integration theory and the fundamentals of global analysis. Once again, emphasis is laid on a modern and clear organization, leading to a well structured and elegant theory and providing the reader with effective means for further development. Thus, for instance, the Bochner-Lebesgue integral is considered with care, since it constitutes an indispensable tool in the modern theory of partial differential equations. Similarly, there is discussion and a proof of a version of Stokes' Theorem that makes ample allowance for the practical needs of mathematicians and theoretical physicists. As in earlier volumes, there are many glimpses of more advanced topics, which serve to give the reader an idea of the importance and power of the theory. These prospective sections also help drill in and clarify the material presented. Numerous examples, concrete calculations, a variety of exercises and a generous number of illustrations make this textbook a reliable guide and companion for the study of analysis