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I'm reading Lang's Real and Functional Analysis, and I am surprised that one can still do a fair amount of calculus (differential/integral) on abstract Banach spaces, not just $\mathbb{R}$ of $\mathbb{R}^N$. For example, Lang writes about Bochner integrals - which is slightly different from the 'usual' Lebesgue integral - which gives you a way to integrate Banach-space-valued maps. Also, he uses theorems of differential calculus (of Banach spaces) to prove results about flows on manifolds, which is quite fundamental to differential geometry.

I'm on chapter 7 right now, and I wonder what other good books are there, dealing with this subject: calculus on Banach spaces. After dealing with integration and differentiation (in that order), Lang moves on to 'functional analysis', but I want to see more applications and examples of calculus; for example, Banach-space-valued power series (on $z\in\mathbb{C}$, say), whether one can use the familiar techniques of complex analysis in that case (e.g. Cauchy integral formula), or how the theory is used for differential topology/geometry.

Can anyone suggest a text that gives a complete/thorough treatment of calculus in Banach spaces? (Ones with geometric flavor are even nicer!) Any advice is welcome.

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  • $\begingroup$ Unsurprisingly, Differential and Riemannian Manifolds by Lang. Dieudonné's Foundations of Modern Analysis and the later volumes of his treatise deal with this subject; however, his treatment of manifolds is finite-dimensional, I believe. $\endgroup$ – user49640 Jun 23 '17 at 10:32
  • $\begingroup$ Concerning your examples, also note that there are two things, which can be replaced by a Banach space, the image of a function and its domain. When you consider functions from $\mathbb{R}$ to a Banach space, as for example with Bochner-integrals, most of the results indeed carry over neatly, often with the same proofs. If you consider functions from a Banach space, even just to $\mathbb{R}$, things get complicated rather quickly. $\endgroup$ – mlk Jun 23 '17 at 11:47
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  1. For a short introduction, I suggest Ambrosetti and Prodi, A primer of nonlinear analysis, Cambridge 1995 (second edition).
  2. Another good source is the Springer book by Abraham, Marsden, Ratiu on Maniflds, tensor analysis and applications.
  3. There is something in J. T. Schwartz's book on Nonlinear functional analysis, Gordon and Breach.
  4. The most complete source is, as far as I know, the book by Cartan, Differential calculus, Hermann.

Anyway, you should keep in mind that differential calculus in normed spaces is rather easy and classical. Integration theory becomes more intriguing and difficult for vector-valued functions.

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http://www.tuloomath.com/my-account/text-cartan-dcalculusonnormspaces-2nd-ed/

You're in luck-Henri Cartan's beautiful Differential Calculus has just been republished in an inexpensive paperback at Createspace. For a generation,it and it's sequel, Differential Forms, have together been considered the definitive text on calculus on Banach spaces. The latter has been available from Dover in an inexpensive paperback for over 10 years now,but without the first half of the course, it's been very difficult to use for course study because Cartan's notation is unique at this level and so it's been hard to find the prerequisites. The first half has been notoriously difficult and expensive to buy.

UNTIL NOW. Click on the link above to be taken to the book's homepage and you'll find out more.

You're welcome.

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