This volume is not part of this online collection.
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.
Readership
Graduate students interested in analysis.