An Epsilon of Room, I: Real Analysis

In 2007 Terry Tao began a mathematical blog to cover a variety of topics, ranging from his own research and other recent developments in mathematics, to lecture notes for his classes, to nontechnical puzzles and expository articles. The first two years of the blog have already been published by the American Mathematical Society. The posts from the third year are being published in two volumes. The present volume consists of a second course in real analysis, together with related material from the blog. The real analysis course assumes some familiarity with general measure theory, as well as fundamental notions from undergraduate analysis. The text then covers more advanced topics in measure theory, notably the Lebesgue-Radon-Nikodym theorem and the Riesz representation theorem, topics in functional analysis, such as Hilbert spaces and Banach spaces, and the study of spaces of distributions and key function spaces, including Lebesgue’s $L^p$ spaces and Sobolev spaces. There is also a discussion of the general theory of the Fourier transform. The second part of the book addresses a number of auxiliary topics, such as Zorn’s lemma, the Carathéodory extension theorem, and the Banach-Tarski paradox. Tao also discusses the epsilon regularisation argument―a fundamental trick from soft analysis, from which the book gets its title. Taken together, the book presents more than enough material for a second graduate course in real analysis. The second volume consists of technical and expository articles on a variety of topics and can be read independently.

Table of Contents

Chapter 1. Real analysis

1.1. A quick review of measure and integration theory

1.2. Signed measures and the Radon-Nikodym-Lebesgue theorem

1.3. Lp spaces

1.4. Hilbert spaces

1.5. Duality and the Hahn-Banach theorem

1.6. A quick review of point-set topology

1.7. The Baire category theorem and its Banach space consequences

1.8. Compactness in topological spaces

1.9. The strong and weak topologies

1.10. Continuous functions on locally compact Hausdorff spaces

1.11. Interpolation of Lp spaces

1.12. The Fourier transform

1.13. Distributions

1.14. Sobolev spaces

1.15. Hausdorff dimension

Chapter 2. Related articles

2.1. An alternate approach to the Carath´eodory extensiontheorem

2.2. Amenability, the ping-pong lemma, and the Banach-Tarski paradox

2.3. The Stone and Loomis-Sikorski representation theorems

2.4. Well-ordered sets, ordinals, and Zorn’s lemma

2.5. Compactification and metrisation

2.6. Hardy’s uncertainty principle

2.7. Create an epsilon of room

2.8. Amenability