Mathematical analysis during the 20th century
This is a superb history of 20th century mathematical analysis. One is taken on a tour of all the major areas with a lot of clear detail and many quotes from those who took part in the developments. NEWLINENEWLINENEWLINEIn the introduction the scope and the definition of analysis is discussed with many quotes, including one by Temple (1981) who says that the only fault of the French analysts is that they did not provide an effective definition of their subject, and one by Penrose (1978) who claims that the picture of physical space led to the basic ideas of continuity and smoothness in analysis.NEWLINENEWLINENEWLINEIn each topic considered the chapters are divided into ``Evolution 1900-1950'' and ``Flashes 1950-2000''. NEWLINENEWLINENEWLINEThe first chapter is General Topology which is fairly thorough but unfortunately, only contains a very small section at the end on Robinson's non-standard analysis that seeks to justify the infinitesimal. We proceed through chapters on Integration and Measure to Functional Analysis. In the latter I was particular interested in the historical development of distributions, although I would have expected some reference to Lighthill's work. The work of Dirac and Schwartz are well covered and there is an interesting comment from the latter to the effect that if he had not found the theory of distributions it would soon have been discovered during this time because of the recent work of other mathematicians. ( This was from a book in 1997). NEWLINENEWLINENEWLINEThe next chapter is on Harmonic Analysis linking to physics through von Neumann's work but also mentioning the classical work of Hardy and Littlewood. Throughout these chapters there are so many topics that one is forced to concentrate on those for which one has a firm interest and I was particularly interested in the next chapter on Lie Groups. Many aspects are covered and inevitably minor errors will occur. I note that on page 181 the last equation should be an expression for \(C\) not for \(\exp(A)\) \(\exp(B)\) which is equal to \(\exp(C)\). NEWLINENEWLINENEWLINEThe next few chapters are written in the same style and cover: The Theory of Functions; Ordinary and Partial Differential Equations; Differential Topology; Probability and Algebraic Geometry. Two topics that I found very interesting were the definition of random variables as defined by Kolmogorov and the clear description by Halmos; that a random variable is a function attached to an experiment and once the experiment has been performed the value of the function is known. The discussion of the Riemann hypothesis is also fascinating. NEWLINENEWLINENEWLINEIn conclusion a book from which most mathematicians would find a lot to interest them.
scientific article; zbMATH DE number 1624444