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The striking theorems showcased in this book are among the most profound results of twentieth-century analysis. The authors' original approach combines rigorous mathematical proofs with commentary on the underlying ideas to provide a rich insight into these landmarks in mathematics. Results ranging from the proof of Littlewood's conjecture to the Banach–Tarski paradox have been selected for their mathematical beauty as well as educative value and historical role. Placing each theorem in historical perspective, the authors paint a coherent picture of modern analysis and its development, whilst maintaining mathematical rigour with the provision of complete proofs, alternative proofs, worked examples, and more than 150 exercises and solution hints. This edition extends the original French edition of 2009 with a new chapter on partitions, including the Hardy–Ramanujan theorem, and a significant expansion of the existing chapter on the Corona problem.
The purpose of this chapter is to bring out appropriate mathematical concepts to express whether a subset of ℝ (to begin with) is small or large. The notion of smallness that we would like to define is subject to three conditions.
(1) Heredity: any subset of a small set must also be small.
(2) Stability under countable union (any countable union of small sets is also small).
(3) No interval [a, b] (with a < b) is small.
A subset of ℝ will be large if its complement is small. If P(x) is an assertion depending on a real number x, we say that P is generic (or typical) if P(x) is true for x belonging to a large subset of ℝ;.
Here, among others, are three possible points of view.
• Cardinality: the small sets are those that are finite or countable.
• Measure: the small sets are those that are negligible in the sense of Lebesgue.
• Category: the small sets are those that are of first category in the sense of Baire.
In what follows, we will leave the first point of view aside, in order to compare the other two notions in a specific situation: the study of the points of continuity of derivative functions. For all this chapter, a good reference is .
This book has a history: it was born after the encounter of two professors from different generations, on the occasion of a series of mathematics seminars organised by the younger of the two at the Lycée Clemenceau in Nantes, in the early part of the years 2000 onwards. The prime objective of these seminars was to allow the professors of this establishment to keep a certain mathematical awareness that the sustained rhythm of preparing students for competitive entrance exams did not always facilitate. The seminars took place roughly once a month, and lasted an hour and a half. Over the years, the professors were joined by an increasing number of students from their classes; a vocation for mathematics was born for many of these, possibly in part due to this initiative. Both authors gave half a dozen talks at these seminars, on themes of their choosing, with a strong emphasis (but not exclusively) on classical analysis.
After the nomination of one of us to Lyon, we thought it would be interesting to assemble and write up these talks in more detail, and to find a connection between them. It seemed to us that a good starting point would be the 1911 paper of Littlewood (Chapter 1), which is at the same time the founding point of what we today call Tauberian theorems, and the beginning of the famous collaboration between Hardy and Littlewood that spanned 35 years, until Hardy's death in 1947. This collaboration produced a large number of remarkable discoveries, not the least of which was that of Ramanujan. The magnificent work of Hardy and Ramanujan on the asymptotic behaviour of the partition function is in fact the subject of an entire chapter (Chapter 8).