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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?

Thanks.

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    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.2011-11-15

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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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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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    @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.2011-09-16
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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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    @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.2011-09-16