Assuming minimal background on the part of students, this text gradually develops the principles of basic real analysis and presents the background necessary to understand applications used in such disciplines as statistics, operations research, and engineering. The text presents the first elementary exposition of the gauge integral and offers a clear and thorough introduction to real numbers, developing topics in n-dimensions, and functions of several variables. Detailed treatments of Lagrange multipliers and the Kuhn-Tucker Theorem are also presented. The text concludes with coverage of important topics in abstract analysis, including the Stone-Weierstrass Theorem and the Banach Contraction Principle.
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13 of 15 people found the following review helpful:
5.0 out of 5 stars
A unique introductory book on real analysis,
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This review is from: Introduction to Real Analysis (Paperback)
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.
1 of 4 people found the following review helpful:
5.0 out of 5 stars
Correcting a mistake in my review...,
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This review is from: Introduction to Real Analysis (Paperback)
In the place where it says that the Feynman-Kac formula is proven using gauge integration, please ignore it, because it's not true.
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In this text, we introduce real analysis. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
tagged division, gauge integrable, gauge integral, seminormed space, converges coordinatewise, semimetric space, intermediate value property, order sufficient condition, absolute integrability, monotone convergence theorem, normed linear space, delta sequence, unbounded intervals, accumulation point, converges pointwise, exponential order, finite subcover, inverse function theorem Key Phrases - Capitalized Phrases (CAPs): (learn more)
Prove Proposition New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
In this text, we introduce real analysis. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
tagged division, gauge integrable, gauge integral, seminormed space, converges coordinatewise, semimetric space, intermediate value property, order sufficient condition, absolute integrability, monotone convergence theorem, normed linear space, delta sequence, unbounded intervals, accumulation point, converges pointwise, exponential order, finite subcover, inverse function theorem Key Phrases - Capitalized Phrases (CAPs): (learn more)
Prove Proposition New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
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14
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- Studies in Modern Analysis by Robert Creighton Buck on page 64, page 320, and Back Matter
- Measure, Integration And Function Spaces by Charles Swartz in Front Matter, and Back Matter
- Aspects of Image Processing and Compression, Volume 119 (Advances in Imaging and Electron Physics) by Peter W. Hawkes on page 191
- Infinite Matrices and the Gliding Hump by Charles Swartz in Back Matter
- Introduction to Gauge Integrals by Charles Swartz in Back Matter
- Advances in Imaging and Electron Physics, Volume 99 by Peter W. Hawkes on page 62
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