From Calculus in $\mathbb{R}^n$ to Calculus in $\mathcal{H}$

Question

Does anybody know of a text (doesn't matter which form - article, book etc. - doesn't matter which language) which makes the transition from analysis in $\mathbb{R}^n$ to analysis in Banach or Hilbert spaces ? (Either space is fine, but both would be a plus.)

I'm thinking of a text that goes through the basic key concept, like the derivative and the integral (Riemann/Lebesgue) and just notes which results from $\mathbb{R}^n$ can be immediately generalized to these abstract settings (in the sense that you can almost replace $\mathbb{R}^n$ everywhere with your abstract space) and which can't. Thus this text should not give a complete exposition of calculus in, say, Banach spaces; it should only serve as to "spot the difference".

On math.SE you can get 100 bounty points if you answer this within the next approx. 20 hours.

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Here's more precise description of what this text should and should not contain, in case there is confusion. For brevity this description shall apply only for the concept of the Lebesgue-integral and the generalization to Hilbert spaces.

Let $L\mathbb{R}$ denote the set of theorems which hold for the L-integral of a function $f:X\rightarrow\mathbb{R}$, where $X$ is a measurable space.
Similarly let $L\mathcal{H}$ denote the set of theorems which hold for L-integrals of functions $f:X\rightarrow\mathcal{H}$, where $\mathcal{H}$ is a, say, Hilbert space.

Now I'm not interested in knowing all theorems from $L\mathcal{H}$, since they probably are rather complicated. I'm only interested in

• those theorems/definitions from $L\mathcal{H}$ whose proof consists only of an "almost mechanical" replacement of all occurences of "$\mathbb{R}$" with "$\mathcal{H}$".

That there may still be some theorem $t_{\mathcal{H}}$ in $L\mathcal{H}$, that is the "appropriate generalization" of $t_{\mathbb{R}}$ from $L\mathbb{R}$, but whose proof and form may divergence significantly from those of $t_{\mathbb{R}}$ and therefore does not interest me.
Additionally I would like to know

• which theorems from $L\mathbb{R}$ fail - or cannot even be formulated - when we replace "$\mathbb{R}$" with "$\mathcal{H}$", i.e. for which theorems this "mechanical" procedure fails.

Answer 1

Banach space is the standard setting for the French Analysis courses of the second half of XX century, for example H. Cartan, Calcul differentiel. Formes differentielles, Herman, Paris, 1967. It does not have Lebesgue integral. The course of Schwartz has everything, including a careful discussion of differences between the finite-dimensional case and the general case. Laurent Schwartz Analyse mathematique I, II. Herman, Paris, 1967. There is also a course of Dieudonne.

Answer 2

Take a look at the long but extremely well written paper ``The inverse function theorem of Nash and Moser '' by Hamilton (in the Bulletin of AMS).