Development of "Classical" Analysis on Metric Spaces and Normed Vector Spaces Most introductory texts on analysis begin by studying the properties of the real line and (either by hypothesis or construction) assert that $\mathbb{R}$ is a complete and totally ordered field. All of analysis then seems to flow from these precepts. On the other hand, metric spaces themselves can be complete and totally ordered and for higher-dimensional developments one can consider Banach spaces. So, other than for motivational and pedagogical reasons, is there any need to actually develop analysis in the context of $\mathbb{R}$/ $\mathbb{R}^n$ instead of complete metric/Banach spaces? Are there results that are unique to $\mathbb{R}$/$\mathbb{R}^n$ that cannot be developed in complete metric/Banach spaces? Of course, everything that is true about these abstract spaces is true about $\mathbb{R}$/ $\mathbb{R}^n$ but does the converse hold?
Update: Since asking this question, I have found a reference that gives a really nice comparison between metric/normed spaces and $\mathbb{R}^n$. It's found on pages 150 - 152 in Marsden's Elementary Classical Analysis (1st Ed). Although the text itself does not develop everything with full generality (hence the title Classical Analysis!), the charts on these pages list the results in the text and indicate whether they hold in abstract spaces and indicates what restrictions need to be placed on a space for a given result to be valid. 
 A: There are tons of results that are specific to $R^n$ and even more that are specific to $R$ (though lots of stuff works on Banach spaces too...)
Generally speaking $R^n$ enjoys the properties of finite dimension : every linear application is continuous (that's not true of the differential in the space $C^{1}$ for example), every bounded closed set is compact (not true at all in infinite dimension and the main motivation for weak topologies), etc.
Also, you have a wonderful measure on those spaces : the Lebesgues measure (it is invariant under isometries, regular, sigma finite, borelian... you name it !). You lose that in infinite dimension, and therefore you lose everything integral-related such as the Fourier transform or distributions.
$R$ has two main additional characteristics : it is a field and it is completely ordered, with the sup property (that's almost as important as completeness). You lose the meaning of monotonic functions for example, as soon as you consider functions defined on anything else than the real line.
What works in Banach spaces is, roughly speaking, differential calculus : differential equations, local inverse theorems, differentials...
Those are only a few examples but I think that your question is essentially what motivates pretty much everything in analysis ;)
EDIT : as for abstract metric spaces, some stuff like convolution requires a structure of group. but I don't mean to say either that nothing can be done in more abstract settings...
A: You might be interested in Loomis and Sternberg's "Advanced Calculus", which does develop as much of this material as possible from a general normed-linear-space point of view.
I'm not necessarily recommending it as a textbook (although in fact it was the text for the first math course I took as an undergraduate at University of Chicago, taught by Max Jodeit).
A: In the comments someone already mentioned Bourbaki. In General Topology they define and develop the necessary theory on $\mathbb{R}$, and also in that book they cover metric spaces (as special case of uniform spaces). In Topological Vector Spaces they cover normed spaces and Banach spaces as special cases.
The downside of this beautiful approach is that it takes a lot of time and effort before you get somewhere. A good alternative is Dieudonne's Treatise on Analysis; especially the first volume. It treats analysis on $\mathbb{R}$ and metric spaces (general, point set topology is done in the second volume), and then general Banach spaces.
(Note that you need some things about $\mathbb{R}$ before metric spaces and normed spaces, since it appears in the very definiton of 'metric' and 'norm'.)
A: A very good-and cheap-analysis text at the Rudin-Apostol-Pugh upper undergraduate level that fully covers the calculus in Banach spaces and general norned spaces is Kenneth Hoffman's Analysis In Euclidean Space. In this book, Hoffman is very careful to show what works in Euclidean space and what doesn't generalize readily to arbitrary normed spaces. In fact,it's the only book I know that develops real analysis on normed spaces only and barely mentions metric or topological spaces! It's a fascinating approach and I think it'll answer a lot of your questions.  
