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**About this book :- **
** An Epsilon of Room, I: Real Analysis ** written by
** Terence Tao **

This text is lecture notes of graduate real analysis courses that are taught at University of California, Los Angeles (Chapter ), together with some related material. These notes cover the second part of the graduate real analysis sequence here, and therefore assume some familiarity with general measure theory (in particular, the construction of Lebesgue measure and the Lebesgue integral), as well as undergraduate real analysis (e.g., various notions of limits and convergence). The notes then cover more advanced topics in measure theory (notably, the Lebesgue-Radon-Nikodym and Riesz representation theorems) as well as a number of topics in functional analysis, such as the theory of Hilbert and Banach spaces, and the study of key function spaces such as the Lebesgue and Sobolev spaces, or spaces of distributions. The general theory of the Fourier transform is also discussed. In addition, a number of auxiliary (but optional) topics, such as Zorn’s lemma, are discussed in Chapter 1.15.2.

**Book Detail :- **
** Title: ** An Epsilon of Room, I: Real Analysis
** Edition: **
** Author(s): ** Terence Tao
** Publisher: **
** Series: **
** Year: **
** Pages: ** 358
** Type: ** PDF
** Language: ** English
** ISBN: **
** Country: ** US

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**About Author :- **

Author ** Terence Tao ** is an Australian mathematician. He was born on 1975 in South Australia. His father was a Chinese paediatricain.

He is a professor of mathematics at the University of California, Los Angeles. His area of research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing, analytic number theory and representation theory. He was awarded the Fields Medal in 2006 and MacArthur Fellowship awarded in 2007.

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** • Download PDF An Introduction to Measure Theory by Terence Tao **

** • Download PDF The Nonlinear Fourier Transform by Terence Tao **

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**Book Contents :- **
** An Epsilon of Room, I: Real Analysis ** written by
** Terence Tao **
cover the following topics.

Preface

A remark on notation

Acknowledgments

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

2. Related articles

2.1. An alternate approach to the Carath´eodory extension theorem

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

Bibliography

Index

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