Analysis (2nd ed.)

Teschl cites this as a good introduction to Lebesgue measure theory. * * * From the preface: "Since the publication of the first edition [[MR1415616](https://mathscinet.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1415616&loc=fromrevtext)] four years ago, we have received many helpful comments from colleagues and students. Not only were typographical errors pointed out, but interesting suggestions were also made for improvements and clarification. We, too, wanted to add more topics which, in the spirit of the book, are hopefully of use to students and practitioners. "This led to a second edition, which contains all the corrections and some fresh items. Chief among these is Chapter 12 in which we explain several topics concerning eigenvalues of the Laplacian and the Schrödinger operator, such as the min-max principle, coherent states, semiclassical approximation and how to use these to get bounds on eigenvalues and sums of eigenvalues. But there are other additions, too, such as more on Sobolev spaces (Chapter 8) including a compactness criterion, and Poincaré, Nash and logarithmic Sobolev inequalities. The latter two are applied to obtain smoothing properties of semigroups. Chapter 1 (Measure and integration) has been supplemented with a discussion of the more usual approach to integration theory using simple functions, and how to make this even simpler by using `really simple functions'. Egorov's theorem has also been added. Several additions were made to Chapter 6 (Distributions) including one about the Yukawa potential. There are, of course, many more exercises as well.'' * * * 1st ed. review: In their preface the authors state that they wrote this unconventional book on analysis for students (and teachers) who "prefer to do something with the material, as it is learned, rather than wait for a full-fledged development of all basic principles''. The first chapter, on measure and integration, leaves many technical (i.e. boring) details to the reader as exercises or to be looked up in one of the many existing books. Chapter 2 treats Lp-spaces without mentioning Banach spaces (Hilbert spaces are introduced very briefly). For instance, the authors say that the fact that the dual of Lp separates elements of Lp is "normally proved with the Hahn-Banach theorem'' but they avoid this by showing that if f≠0, then g=df|f|p−2f ̄ ̄ ̄∈Lp′ and ∫fg≠0. Similarly, they prove Mazur's theorem, according to which if fj→f in Lp (1