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][1] 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.


  [1]: http://math.stackexchange.com/questions/541512/from-analysis-of-realvalued-functions-to-analysis-of-hilbert-banach-valued-funct