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.
Best Answer
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.