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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.

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A metric space includes the data of a function $d : M \times M \to \mathbb{R}$. How do you prove things about metric spaces without knowing anything about $\mathbb{R}$? – Qiaochu Yuan Nov 22 '11 at 19:48
@QiaochuYuan Yes, I understand that, but many texts seem to develop a lot of machinery specifically for $\mathbb{R}$ that, as far as I can tell, could be done almost as easily using metric/vector spaces. Given that, the question is just posing what happens only in $\mathbb{R}$ but doesn't happen in metric/vector spaces – ItsNotObvious Nov 22 '11 at 20:26
I actually don't know what machinery you could be talking about (e.g. I don't think Rudin does what you describe). Can you give examples? – Qiaochu Yuan Nov 22 '11 at 20:33
up vote 2 down vote accepted

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...

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Just a quibble about "the differential in the space $C^1$". The derivative of a $C^1$ function is only continuous, not $C^1$. I think what you mean is the derivative from $C^1[0,1]$ (with the topology of $C[0,1]$) to $C[0,1]$. Of course $C^1[0,1]$ with that topology is not a Banach space. To get examples of a discontinuous linear map from Banach space $X$ to Banach space $Y$, you need to use the Axiom of Choice: see – Robert Israel Nov 22 '11 at 19:01
I'm unsure about some of your points; for example, in $\mathbb{R}$ isn't the least upper bound property equivalent to the completeness property (where a sequence converges iff it is Cauchy)? And, if metric spaces can be complete (which they can) how, in this instance is $\mathbb{R}$ special? – ItsNotObvious Nov 22 '11 at 19:05
Also, with regard to dimensionality, since $\mathbb{R}^n$ is a finite dimensional vector space of dimension $n$, I don't see how the consideration of infinite dimensional vector spaces are relevant to this question. – ItsNotObvious Nov 22 '11 at 19:15
@RobertIsrael you are right, I meant that discontinuous linear maps are frequent when you don't require that they be defined everywhere (a phenomenon that does not arise in finite dimension) 3Sphere: those 2 properties are true for R, so it makes little sense to say they are equivalent (it is not completely trivial to prove one using the other). also do you have any example of reasonnably natural totally ordered complete metric spaces with the least upper bound property besides R ? I very much doubt so... – Glougloubarbaki Nov 22 '11 at 19:16
@3Sphere "Are there results that are unique to R/Rn that cannot be developed in complete metric/Banach spaces?" The point is, you are sometimes led to consider maps between Banach spaces, but an analytic treatment of those is necessarily very different from the finite dimension case (off the top of my head, how would you define partial derivative for a non linear map from $L^2$ to itself ? – Glougloubarbaki Nov 22 '11 at 19:22

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).

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well, if you like maximal generality settings, nothing beats bourbaki =) – Glougloubarbaki Nov 22 '11 at 19:34
I actually like that text very much, aside from a few of the choices in notation. And, from what I recall, they do indeed develop most of analysis in abstract spaces. So, if I understand your point, they only develop in $\mathbb{R}$ what must be developed in $\mathbb{R}$ and therefore this text could effectively provide a good answer to my question? – ItsNotObvious Nov 22 '11 at 19:36
I myself am not so familiar with bourbaki books, but the whole purpose of the bourbaki group was precisely to develop all mathematics in a context as rigorous, abstract and general as possible (some say, at the cost of intuition and clarity). kind of the exact opposite of arnold, if you read any of his books. so yes, if you want optimal settings bourbaki should be the place to look. – Glougloubarbaki Nov 22 '11 at 19:43
for example, I don't think they shy away from any of the technicalities involved in topics like distributions and topological vector spaces, as too many authors do. – Glougloubarbaki Nov 22 '11 at 19:45

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'.)

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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.

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