An Epsilon of Room, I: Real Analysis

Description

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. Contents Preface 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 Caratheodory 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