Real analysis: modern techniques and their applications
An in-depth look at real analysis and its applications-now expanded and revised.
This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory.
This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include:
* Revised material on the n-dimensional Lebesgue integral.
* An improved proof of Tychonoff's theorem.
* Expanded material on Fourier analysis.
* A newly written chapter devoted to distributions and differential equations.
* Updated material on Hausdorff dimension and fractal dimension.
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Folland's proofs are straight from 'The Book', the elegance is legendary. Royden (not Rudin) has an analysis book that attempts this elegance but fails in comparison. The steps stripped from Royden to shorten his proofs are essential to the proof, and a novice student can do themselves great harm by learning from his book without a guide. Folland's omissions are brilliant, they are not as subtle so they beckon the reader to explore, but yet their omission is safe, because the proof-as-is contains all the core details and is never made misleading by the omissions. This is hard to explain , or at least I am having difficulty.
Your maturity is a factor here, examples must be created yourself, you must constantly test your knowledge by comparing it to examples and seeing if the outcome is proper. Every detail is important, every letter. You will miss a detail, you must devise a way to test your knowledge to catch these details. After the ultra-abstract foundations are built, the functional analysis sections FLOW LIKE BUTTER. All the work in the earlier sections pays off when the power of the abstract tools is used with full strength and dexterity, I love it.
I've read from Rudin and Royden and Kolmagorov and a few more. They all have their pros and cons, some better than others (Kolmagorov is a stand-out). But none of them can capture the magic behind Folland's book. You will never see writing quite like it, and after battling with his text (which will be done slowly I assure you, 2 pages folland are 20 in another text) you will learn to love it like an old friend. Also, the more compact a book, the better a reference it is, so one should at least keep it on the shelf.
To a student of measure theory / functional analysis:
All of the abstractions were motivated by real problems at the onset of the last century. Mathematics underwent it's own revolution , similar in ways to that of quantum mechanics /relativity in physics, and other similar revolutions in art, war, philosophy...The point is that the brutal and beautiful abstractions are not historically present because some old mathematician hates you, it's because difficult problems required new mechanics, that Riemann's old ways couldn't handle. It's at least worth finding the reason so much effort went into the creation of measure theory, and what it can do for a mathematician.
My advice is to compare Atkin's numerical analysis with the functional analysis version he has for numerical analysis. The power shines in the difference from these two books. Or look at probability