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**Real analysis.**
*(English)*
Zbl 0997.26003

Cambridge: Cambridge University Press. xiii, 401 p. (2000).

It is a textbook on real analysis at the master level, thought for an heterogeneous audience (students of pure and applied mathematics, statistics, engineering, economics, education) with different backgrounds. The challenge is to propose a self-contained course which can offer something of value to both specialists and non-specialists; not too pedestrian for the more experted students, not too hard for the less well-prepared ones.

The book is divided in three parts: metric and normed spaces (as a beginner’s guide to general topology), spaces of functions (as a transition from abstract spaces to the “concrete” space of continuous real-valued functions), Lebesgue measure and integration (on the real line). Even if a background on advanced calculus is assumed, a review of the main topics is recalled in order to get the course self-contained. The material is presented in a conversational style which offers many hints for reflection and study in depth, some details are left to the students for encouraging an active learning. The main ideas are introduced in several different forms before their rigorous definition. The “utility” of new concepts is emphasized. A section of notes and remarks – at the end of each chapter – contains additional topics, alternate presentations, historical commentary and references to original works. A great variety of exercises are proposed through the text. They are considered as a part of the presentation and those more important for a full understanding of the material are marked with a small triangle. A rich list of references and two useful indexes (symbol and topic) are included.

The book is divided in three parts: metric and normed spaces (as a beginner’s guide to general topology), spaces of functions (as a transition from abstract spaces to the “concrete” space of continuous real-valued functions), Lebesgue measure and integration (on the real line). Even if a background on advanced calculus is assumed, a review of the main topics is recalled in order to get the course self-contained. The material is presented in a conversational style which offers many hints for reflection and study in depth, some details are left to the students for encouraging an active learning. The main ideas are introduced in several different forms before their rigorous definition. The “utility” of new concepts is emphasized. A section of notes and remarks – at the end of each chapter – contains additional topics, alternate presentations, historical commentary and references to original works. A great variety of exercises are proposed through the text. They are considered as a part of the presentation and those more important for a full understanding of the material are marked with a small triangle. A rich list of references and two useful indexes (symbol and topic) are included.

Reviewer: A.Salvadori (Perugia)

### MSC:

26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |