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I'm looking for a book that gives an overview of analysis, a bit like Shafarevich's Basic Notions of Algebra but for analysis. The book I have in mind would give definitions, theorems, examples, and sometimes sketches of proofs. It would cover a broad swathe of analysis (real, complex, functional, differential equations) and discuss a range of applications (i.e. in physics and in prime numbers). I've looked at the Analysis I volume of the Encyclopaedia of Mathematical Sciences which Shafarevich's book is also a part of, but it focuses more on methods and isn't quite what I have in mind.

Thank you!

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Have you looked at the Princeton Companion to Mathematics? –  Potato Oct 27 '13 at 17:34
    
Is there an Analysis volume in the Schaum's Outline series? If so, it might be just what you're looking for. –  MJD Jan 15 at 5:28
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6 Answers 6

Loomis & Sternber's Advanced Calculus is available online. It is a classic that goes well beyond what people normally call calculus (differential equations, differential geometry, variational principles, ...). Personally I really like Sternberg's books, but it is a full blown textbook rather than a survey.

Or maybe Aleksander & Kolmogorov, Mathematics: Its Content, Methods and Meaning is closer to what you are looking for. It is more of a survey, but with a lot of depth. It is much broader than what you asked, but it is 1100+ pages and it is biased towards analysis related topics.

Another area to explore are advanced applied mathematics texts, these are often primarily analysis oriented, comprehensive and, well, application oriented and you can find books by people like Kreyszig, Lanczos, ... Dover have published a number of books like this.

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You can try, for example, read "Elements of the Theory of Functions and Functional Analysis" A. N. Kolmogorov, S. V. Fomin.

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These are nice books but again, not exactly what I am looking for. –  Marius Kempe Jan 13 at 7:10
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There is a series of books titled Princeton Lectures in Analysis that i'd recommend; it's a 3 volume series covering Fourier, complex and real analysis. They're considered introductions, but I would hardly consider them that; it assumes you are already familiar with basic mathematical analysis. I own the first volume (Fourier analysis) and it's surpassed my expectations - there are quite a lot of applications (entire chapters devoted specifically to certain applications, like BVPs and the Heisenberg Uncertainty Principle). To boot, it provides a bit of historical context, which is always nice. The only downside is that, after a few examples, the author starts skipping steps in the remaining examples - it can be a bit difficult to follow, but if you work through the example yourself, you can fill in the gaps.

Here's a link to the Fourier analysis book:

http://www.amazon.com/Fourier-Analysis-Introduction-Princeton-Lectures/dp/069111384X

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This book might not be exactly what you want (since you just want an overview) but I believe its a really good deal specially because its only 13 bucks and its suppose to be a gentle introduction book for analysis. Also, its author is an experienced mathematician who as taught Analysis for soooo many year at the Massachusetts Institute of Technology.

Also, it is very good because it covers everything from the ground up in proofs, so if you are unfamiliar with proofs, it gives an overview of the different proof methods (direct, indirect and other stuff). The book also gives me the impression it has examples in every chapter, which is always useful. Thought, I believe its not as broad as other books because it suppose to be an intro book.

Also, it has an online companion for the full course on http://ocw.mit.edu/courses/mathematics/18-100a-introduction-to-analysis-fall-2012/

Book at:

http://www.amazon.com/Introduction-Analysis-Arthur-Mattuck/dp/1484814118

Hope this helps!

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I am not looking for an introduction to analysis - there are hundreds of those. –  Marius Kempe Jan 13 at 7:05
    
Sorry, I was just trying to help... also, its really cheap and from an experienced prof. I thought it might have helped you. If its really that unhelpful for you I can delete the post. –  Charlie Parker Jan 13 at 7:06
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Another three volume set "A Course in Mathematical Analysis". Good set of introductory books with results/theory/proof but also intuition, examples and even exercise. Also contains applications.

http://www.amazon.com/Course-Mathematical-Analysis-Volume-Set/dp/1107625343/

Go to the first volume to look at table of contents for the entire set (on Amazon). Probably the best in a single "item" along with the Princeton series. (Ignore the rating on Amazon - it is due to a poor Kindle version)..

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I should note I use the term introduction also but I guess that is a subjective term.. certainly it is detailed to the level of proofs.. I would also note that I am not familiar with too many books that give only sketch proofs regularly - I like this set because it follows with intuition.. –  John U Jan 13 at 9:45
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It is impossible to cover analysis in one volume, or even in several. However, Rudin's Principles of Mathematical Analysis, Real and Complex Analysis, and Functional Analysis, read in that order, give you a good overview of a large part of analysis.

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