Herbert A., Joachim E. Analysis III. 2009

Textbook in PDF format

This third volume concludes our introduction to analysis, wherein 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.
This book is also suitable as background for other courses or for self study. 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.
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Table of Contents:
Chapter IX Elements of measure theory
1 Measurable spaces
2 Measures
3 Outer measures
4 Measurable sets
5 The Lebesgue measure
Chapter X Integration theory
1 Measurable functions
2 Integrable functions
3 Convergence theorems
4 Lebesgue spaces
5 The n-dimensional Bochner–Lebesgue integral
6 Fubini’s theorem
7 The convolution
8 The substitution rule
9 The Fourier transform
Chapter XI Manifolds and differential forms
1 Submanifolds
2 Multilinear algebra
3 The local theory of differential forms
4 Vector fields and differential forms
5 Riemannian metrics
6 Vector analysis
Chapter XII Integration on manifolds
1 Volume measure
2 Integration of differential forms
3 Stokes’s theorem