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Can anyone suggest a text that gives a complete/thorough treatment of calculus in Banach spaces? Perhaps something something along the lines of Chapter 2 in Manifolds, Tensor Analysis and Applications by Marsen, et. al. but more expansive?


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Look for texts named "nonlinear analysis" (or something similar) in their name. You're bound to find something you like. – Mark Sep 16 '11 at 12:33
Related to Mark Schwarzmann's comment: many books on PDE contain material similar to this. The words Bochner integral and Frechet derivative may be useful for searches. – user12014 Nov 15 '11 at 21:12

The only one I know is Jean Dieudonne's classic Foundations of Modern Analysis, which is the also the most cited out of print text I know. It's kind of strange since in the late 1960's and early 1970's, at the height of the Bourbaki Era, it was the single most cited analysis text for undergraduates and was fully expected to replace Rudin as the gold standard.

Anyway,I think you'll find exactly what you want in that text-if you're ready for it. It's a tough read-get your pencil and paper ready when you go to read it! there's now an inexpensive edition.

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This book is the first volume in Dieudonné's Treatise on analysis, which has many volumes. I've never browsed the whole thing, and I don't know if the first volume has what would be called a "thorough treatment", but I agree it's worth checking out Chapter VIII. – Jonas Meyer Sep 15 '11 at 19:12
For the part of classical analysis we generally call calculus on Banach spaces,yes,the first volume of Dieudonne's treatise is pretty complete,Jonas.The later volumes are considerably more advanced-and beyond volume 4,they get fairly specialized as well. – Mathemagician1234 Sep 15 '11 at 19:35
Thanks for the info. I suspect that only 3Sphere can verify whether it is complete/thorough to his or her liking. I have some doubts due to there being only one chapter directly on the topic, which is much shorter than the chapter referred to in the question. – Jonas Meyer Sep 15 '11 at 19:42
I'm not sure if there is such a book in the common literature,3Sphere. The wonderful book ANALYSIS IN EUCLIDEAN SPACE by Kenneth Hoffman-now in Dover and reviewed by me for the Mathematical Association of America Online-has an entire chapter on how calculus behaves in Banach spaces. It's only a chapter,but it may be exactly what you're looking for.Also, Serge Lang's DIFFERENTIABLE MANIFOLDS and its' many alternate versions have brief and abstract treatments of differential calculus in Banach spaces.Lastly-Cartan's classic DIFFERENTIAL CALCULUS has a discussion of calculus in Banach spaces. – Mathemagician1234 Sep 16 '11 at 3:11
@Theo Apology accepted. In my younger days,believe me,my response wouldn't have been so civil. I've worked very hard under some IMPOSSIBLE conditions (a father dying of a protracted cancer illness,then my own illnesses). I don't take kindly to someone politely calling me an imbecile because I dared to make an error on a comment when I wasn't clear minded enough (2 hours sleep and up 20 hours) to properly think it through. But as I said,apology accepted and let's move on. – Mathemagician1234 Sep 16 '11 at 6:12

I studied it from Cartan's book: Differential Calculus. I found it a good exposition of the theory. That's my indication :)

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Unfortunately,that book is out of print and VERY hard to locate any copy,let alone an affordable one. Why Dover never bought the rights and reissued it and it's sequel, DIFFERENTIAL FORMS, in a single volume is beyond me. – Mathemagician1234 Feb 6 at 6:27

Bourbaki's "Functions of one real variable" would do the job. It is quite expansive. You could even work in conjuction with with the Bourbaki volumes on Topology, and Topological vector spaces. But the pedagogical value is as always questionable.

Edit: If your aim is to learn analysis on Banach manifolds, then Serge Lang's book on differentiable manifolds is the answer.

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These books may be too advanced for 3Sphere's purposes,George. That's the only problem-otherwise, there's a traincar full of good books on analysis in Banach spaces I could recommend. – Mathemagician1234 Sep 16 '11 at 17:50
@Andrew: The OP wanted a book detailing calculus on Banach spaces. If he had simply asked for an introductory book on Banach spaces, I would have mentioned some simpler book like G. F. Simmons or Rudin. But learning calculus on such generality as encompassing Banach Spaces -- that precisely is what Bourbaki is fond of. – George Sep 16 '11 at 17:54

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