This book is a somewhat extended version of the author's previous [“Analyse fonctionnelle élémentaire” (Enseignement des Mathématiques (Cassini) 17; Paris: Cassini) (2003; Zbl 1089.46001)], and much of what was said about that title refers to the monograph under review as well. Like its predecessor, this interesting book gives a concise introduction to functional analysis and integration theory with a view towards applications to PDEs.

After a chapter on metric spaces, the author deals with integration for which he uses an elementary Daniell-Stone type approach. The next chapters cover the basics of Banach and Hilbert space theory, with special emphasis on $L_p$-spaces and their duality. In particular, the Hahn-Banach extension theorem is proved constructively for separable spaces and for smooth and uniformly convex spaces, but no comment is made about the general case. Also, the Banach-Steinhaus theorem is proved, but the open mapping and closed graph theorems do not appear.

In the second half of the book, more specialised topics are presented. The chapter on Sobolev spaces treats, among other things, the calculus of Sobolev functions, embedding and compactness theorems, trace theorems and extension theorems. Another chapter deals with the notion of capacity and multivariate functions of bounded variation. A further chapter discusses the Dirichlet problem and eigenfunctions of the Laplace operator. This chapter also contains recent results on the symmetrisation of functions leading to the isoperimetric inequality in the setting of multivariate functions of bounded variation. The final chapter presents historical notes, including a brief introduction to the theory of distributions.

The organisation of the material is very original, and so are many of the exercises that accompany each chapter. For example, a key property of $L_p$-spaces on which the author bases his approach is their uniform convexity that is proved in a rather unconventional manner, building on a bivariate Jensen type inequality. Also the smoothness of $L_p$-spaces is crucial, and the Hahn-Banach extension theorem is then proved in this setting, following ideas of R. C. James. In the section on duality of Hilbert spaces, the author first proves the Fréchet-Riesz theorem by a differentiability argument and then derives the existence of orthogonal complements. There are many other noteworthy arguments in this fine book.

The author's style and notation are precise and concise, indeed the language is concise to the point of being minimalistic. The rather dry Definition-Theorem-Proof pattern without barely any commenting or motivating text prevails.

In spite of this criticism, this is a very original contribution to the textbook literature.