Customer Reviews

Introduction to Real Analysis

5 star:		(2)
4 star:		(0)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

So, we have another book on introductory real analysis. Yes, But this one has its own shine, keeping it apart from the classics (Rudin, Apostol, Bartle): It contains the first (and maybe the only) elementary exposition of the marvellous gauge integral. It's a clever and surprising extension of the definition of the Riemann integral, made by Kurzweil and Henstock in the fifties, with the purpose of constructing a Riemann-type integral with all the properties of the Lebesgue integral, and recovering the original spirit of Newton and Leibiniz (as far as primitives are concerned). McShane later showed that this kind of integral totally supersedes the concept of Lebesgue integral and measure. The gauge integral also supersedes the improper and Stieljes integrals, with the big pedagogical advantages of the Riemann integral.

However, the book form presentations of this integral (given mainly by its creators) are incomprehensible, period. On the other hand, Depree/Swartz achieves the previously unfulfilled purpose of giving a human-readable presentation of the gauge integral. It uses it as the main tool for teaching integration, and this is great, because all the book is very readable and down-to-earth. This book has other pearls, like his great presentation of differentiation in several variables (done with Frechet and Gateaux derivatives in a very smooth and clear way, better than Lang), and good topological stuff concerned with analysis as needed. It develops a different way of thinking about analysis, and all you need is a little basic calculus.

All things concerned, it's a first-class book that deserves to be read more and more. Gauge integration is a unfairly forgotten tool, that can enlighten many unsolved problems in mathematics (constructive mathematical analysis) and physics (path integration, as seen for example in the Feynman-Kac formula, proven using gauge integration), or simply make people think about advanced integration and analysis in a simpler way.

In the place where it says that the Feynman-Kac formula is proven using gauge integration, please ignore it, because it's not true.