Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ?
I'm thinking of a text that goes through a basic introductory book (the more known it is, the better) on real analysis - like Principles of Mathematical Analysis by Rudin - and explains which theorems from real analysis still hold in the setting of Hilbert spaces and which will fail. (A discussion for Banach spaces would also be ok - and I'm also interested in this case - but right now the case for Hilbert spaces is more pressing)
To be more precise, I would like to know which results from chapters 5 (Differentiation), 6 (The Riemann-Stieltjes Integral), 7 (Sequences and Series of Functions), 9 (Functions of Several Variables) and 11 (The Lebesgue Integral) generalize as just described.
Of course Rudins book was just an example of a book which I used, because the book is very known. Any solid (but not-too-graduate-level) analysis book, which treats functions $\mathbb{R}\rightarrow \mathbb{R}$ and $\mathbb{R}^n\rightarrow \mathbb{R}^m$, has chapters like those which cover pretty much the same basic theorems.
To give an example of what I already know: The generalization of chapter 4 to Hilbert spaces I know already. In functional analysis the concept of continuity is extended from functions $\mathbb{R}\rightarrow \mathbb{R}$ to mappings $\mathcal{H}\rightarrow \mathcal{H}$ (or more general: metric spaces).
Which results can be extended to which setting ($\mathcal{H}\rightarrow \mathcal{H}$ or $\mathbb{R}\rightarrow \mathcal{H}$) depends on the concept of course: Derivatives can be extended to the first case, integrals only to the second (as far as I know)).
To give a further example: A lot of theorem for derivatives between Hilbert/Banach spaces have a proof that is the same as the proof of the theorem in the case $\mathbb{R}\rightarrow \mathbb{R}$. I'm interested exactly in these theorems -- and want to know which theorems (maybe with the helpt of a counterexample, if the theorem would even make sense in the abstract setting) can't be adapted like this to the more abstract setting (what the "right version" of that $\mathbb{R}\rightarrow \mathbb{R}$ theorem - that fails when "mechanically" extended to $\mathcal{H}\rightarrow \mathcal{H}$ - would be in the abstract setting doesn't interest me at this point).
(I know of course that there are books that develop the whole analysis for Hilbert/Banach spaces in the first place - most graduate analysis books seem to do that - but I don't want to go through the whole book and look at every proof to see if it goes through in the case of Hilbert/Banach spaces)
EDIT: Here's more precise description of what I want, in case there is confusion. I shall describe what I'm looking for for brevity only for the concept of the Lebesgue-integral. But the same holds for all of the other basic concepts in analysis (like the derivative of functions $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$, all of the topological properties of $\mathbb{R}^n$ etc.).
Let $L\mathbb{K}$ denote the set of theorems which hold for the L-integral of a function $f:X\rightarrow\mathbb{K}$, where $\mathbb{K}=\mathbb{R},\mathbb{C}$ and $X$ 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 a "mechanical" replacement of all occurences of "$\mathbb{K}$" with "$\mathcal{H}$", i.e. the theorems from $L\mathbb{K}$ which generalize immediatly to $L\mathcal{H}$.
That there may still be some theorem $t_{\mathcal{H}}$ in $L\mathcal{H}$, that is the "appropriate generalization" of $t_{\mathbb{K}}$ from $L\mathbb{K}$, but whose proof and form divergence significantly from those of $t_{\mathbb{K}}$ does not interest me.
Additionally I would like to know which theorems from $L\mathbb{K}$ fail - or cannot even be formulated - when we replace "$\mathbb{K}$" with "$\mathcal{H}$", i.e. which fail when "straightforward generalized".
