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About Mathematical analysis (University mathematical texts)
This book assumes a standard first course in analysis and gives a unified treatmentof several topics taught in the last two years of an honours degree course In the first chapter the standard topological notions are introduced in a metric space setting complete metric spaces are defined and the contraction mapping theorem and Baire s theorem proved The second chapter gives the standard properties of continuousfunctions between metric spaces pointwise convergence of functions is studiedand its shortcomings motivate a discussion of uniform convergence among thetheorems proved are Dini s the Stone Weierstrass and the Ascoli Arzela Chapter 3 gives further results on uniform convergence in particular its relationship to thepreservation of Riemann integrability and differentiability the former establishingthe need for a more general integral leads naturally to Chapter 4 Here the Daniel extension procedure is applied to the lattice of continuous functions of compact supportand the Riemann integral the use of Dini s theorem makes this particularly neat The relationship to the measure theoretic approach is clearly explained Finally double integrals are introduced and the Fubini and Tonelli theorems proved herethe extension procedure is applied to a lattice of step functions to avoid assumingknowledge of Riemann integration on the plane in the reviewer s opinion it would havebeen preferable to take the continuous functions of compact support and therepeated integral obtaining the equality of the two repeated integrals as an applicationof the Stone Weierstrass theorem The last chapter gives the L1 and L2 theories ofthe Fourier transform This book is very attractive the treatment is leisurely and while it is consistentlyrigorous many of the harder theorems have their proofs supplemented by informalexplanations and graphical illustrations also a number of interesting applicationsare given Each chapter has a large number of interesting exercises some of whichextend the theory while others give relevant counter examples Although the reluctance of undergraduates to buy books is to be deplored it must be reckoned with The low priceof this book its suitability for several courses and the high standard ofits presentation combine to make it ideal as a course book FREDA E ALEXANDER
Detailed Information
| Author: | Gordon H. Fullerton |
|---|---|
| Publication Year: | 1971 |
| ISBN: | 9780050023464 |
| Pages: | 158 |
| Language: | English |
| File Size: | 4.1 |
| Format: | |
| Price: | FREE |
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