Why study metric spaces?
Most universities have a 3rd year undergraduate analysis course in which metric spaces are studied in depth (compactness, completeness, connectedness, etc...). However, in practice it seems that most of these metric spaces are normed vector spaces. Why not just cover normed vector spaces instead of metric spaces?
Even if we lose some generality, normed vector spaces feel more natural and interesting, in my opinion, at least.
real-analysis metric-spaces education
Martin Sleziak
nigelnigel
$\begingroup$ By 'normed space' do you mean 'normed vector space'? $\endgroup$
– Michael Albanese
$\begingroup$ @MichaelAlbanese: Yes. I'll edit the question. $\endgroup$
– nigel
$\begingroup$ @nigelvr : good question. What would be the point of giving up the generality? I don't think the basic topological properties of normed vector spaces and subspaces thereof would be that much easier to prove than those of arbitrary metric spaces, if that's what you're after. $\endgroup$
– Stefan Smith
$\begingroup$ @StefanSmith: My ignorance is probably showing when I say this, but I don't understand the point of studying metric spaces in such detail when they don't come up very much in practice. While studying normed vector spaces wouldn't necessarily be easier, it would be more motivated. $\endgroup$
$\begingroup$ @nigelvr : if you proved things only for subspaces of normed vector spaces first, it might be more motivated, but you would have to prove everything over again for the general case, a huge waste of time. And presumably in a real analysis class, you want to prove things rather than take them on faith. Plus a good teacher can use examples from familiar spaces like $\mathbb{R}$ to illustrate ideas. $\endgroup$
You write in a comment that metric spaces "don't come up very much in practice". This is not true (although may reflect the mathematics you have seen so far).
Metric spaces (and, more generally, topological spaces) occur all over the place. I am a working number theorist, and I use the concepts of topology (in all kinds of contexts, sometimes in the context of vector spaces or rings or groups, sometimes in very non-linear contexts) all the time. Geometers and topologists use them still more frequently (perhaps unsurprisingly).
The language and basic results of topology (open and closed sets, continuity, connectedness, compactness) are some of the most flexible and useful concepts in mathematics!
Added: In my answer I've tended to conflate metric spaces and general topological spaces, but perhaps in your question you want to distinguish them. (Maybe you are wondering where particular metrics arise that are not induced by normed spaces.)
To the extent that geometry is about studying lengths, angles, and related concepts such as curvature, it is very much a subject that revolves around metric spaces, and in modern geometry, geometric topology, geometric group theory, and related topics, many techniques use metrics as the basic strucure.
E.g. Gromov--Hausdorff limits.
E.g. Metric space approach to concepts such as curvature, leading e.g. to CAT-0 spaces.
E.g. You might be tempted to think as Riemannian geometry as being a more analytic than combinatorial/metric-space based subject, because of the role of differential topology in the foundations. But metric space notions (such as the two previous examples) are fundamental in modern aspects of the theory, such as rigidity.
Matt EMatt E
$\begingroup$ Perhaps I wasn't clear. What I meant was that most metrics are induced by "fairly natural" norms. I also never said I had a problem with topological spaces, though topological spaces, for me, were not introduced in real analysis, but in differential geometry. edit: this wasn't meant to be confrontational, and I acknowledge that I am fairly ignorant. $\endgroup$
$\begingroup$ @nigelvr : FWIW : you don't sound confrontational to me. $\endgroup$
$\begingroup$ Dear nigelvr, But your statement about metrics is not true. E.g. if I take a Riemannian manifold, how is the metric on it induced by a norm? If take a tree, how is the path-length metric induced by a norm? When I take the $I$-adic completion of a ring $R$, how is the $I$-adic metric induced by a norm? (These are just a few examples that came to mind.) Regards, $\endgroup$
– Matt E
$\begingroup$ @nigelvr: Dear nigelvr, You don't sound confrontational, and I don't mean to sound so by contradicting you either. It's just that if all metric spaces in nature arose as subsets of normed spaces, your point of view would make sense; I'm trying to give some examples showing why this isn't true, and hence explain why people don't share your view point! Best wishes, $\endgroup$
$\begingroup$ I was just thinking about Riemannian manifolds when I posted my question! I was thinking that the metric one has on a Riemannian manifold (obtained by taking infima of lengths of paths from $a$ to $b$). To me that doesn't seem like the most exciting metric (topologically speaking). I was also thinking of hamming weight. But given that people here seem to be interested in such metrics, I guess there is indeed real interest. So, I suppose it is fairly well motivated. edit: $I$-adic metric is a good example. Thanks for the helpful comments. $\endgroup$
Metric spaces are far more general than normed spaces. The metric structure in a normed space is very special and possesses many properties that general metric spaces do not necessarily have.
Metric spaces are also a kind of a bridge between real analysis and general topology. With every metric space there is associated a topology that precisely captures the notion of continuity for the given metric. That means that many topological properties can be understood in the context of metric spaces, encompassing many many examples of varying degrees of complexity, while still having a notion of distance. The notion of distance is typically seen as much less abstract than that of a topology, with many of the proofs just being a re-formulation of the standard proofs from real analysis. In fact, somewhat less known is that a naturally occurring generalization of metric spaces (i.e., allowing the metric to attain values not just in the range $[0,\infty]$ but rather in what is known as a value quantale) is as general as topological spaces. That is, every topological space arises as the associated topology for some metric structure in that sense.
Also, many universities are somewhat reluctant to introduce abstract notions early on and so even though the limits in real analysis can be taught just as special cases of metric analysis, and thus introduce metric spaces in the first year, universities often choose to postpone metric spaces. However, this is not always the case as I had the opportunity to teach metric spaces in a first year course (at Utrecht University).
So, this is largely a question of tradition and certainly varies between universities. As is commonly the case, university curricula tend to follow historical developments and tend to adapt and change rather slowly. Metric spaces were introduced in 1906 by Frechet and thus are quite newer than the traditional primary objects of study such as $\mathbb R$ and various function spaces. This may explain why the latter are often taught first.
Ittay WeissIttay Weiss
$\begingroup$ I find it amazing that the simple idea of metric space was not codified until 1906 after people had been doing very difficult work with function spaces for a long time. I just taught a real analysis class. I taught the abstract metric space material before the "functional analysis" material. It worked fine. The students actually did better on the basic metric space material. Our textbook followed the approach you describe and awkwardly used ideas of compactness and connectivity, working with function spaces, before defining them properly later in the book. $\endgroup$
$\begingroup$ In a linear algebra class, I follow the standard approach of doing some matrix algebra before introducing abstract metric spaces, subspaces, and linear transformations, but in a real analysis class, proof is a greater part of the class, and the students are going to be proving things all semester $\endgroup$
$\begingroup$ @StefanSmith I find it more amazing that the concept of metric space was not fully axiomatized until some 80 years later. By that I mean that Frechet's axiomatization introduced the abstract notion of a metric but still taking values in the very concrete interval $[0,\infty ]$. A full axiomatization would also distill only those properties of $[0,\infty ]$ that are really necessary for the metric machinery to work. $\endgroup$
– Ittay Weiss
$\begingroup$ can you please provide a reference for generalized "metrics" that take values that are not in $[0,\infty]$? Is some other ordered field involved? $\endgroup$
$\begingroup$ @StefanSmith not an ordered field (since really the field axioms are far too strong for the purposes of the metric machinery) but rather a value quantale. Flagg's "quantales and continuity spaces", algebra universalis, is where the axiomatization I refer to is established. I also have a note on the arxig titled "A note on the metrizability of spaces" which shows Flagg's construction as a sort of dual solution to the metrization problem to the Bing-Nagata-Smirnoff theorem. $\endgroup$
Metric spaces are more general than normed spaces, because they need not be vector spaces. They are easier than general topological spaces, but introduce all of the relevant concepts.
Igor RivinIgor Rivin
Another perspective might be useful for others looking at this question. Studying subsets of normed spaces is essentially the same as studying metric spaces. Given a metric space $(M,d)$, we can define a vector space $L^\infty(M)$ of bounded functions $f: M \to \mathbf{R}$ with domain $M$. Given a point $x \in M$, consider the function $f_x$ given by $f_x(y) = d(x,y)$. Then for any pair $x_1, x_2 \in X$, $f_{x_1} - f_{x_2} \in L^\infty(M)$, and moreover, $d(x_1,x_2) = \| f_{x_1} - f_{x_2} \|_{L^\infty(M)}$. Thus if we fix $a \in M$, and consider the map $F: M \to L^\infty(M)$ given by $F(x) = f_x - f_a$, then this identified $M$ with the subset $F(M)$ of $L^\infty(M)$. So it is natural that most metric spaces you can think of can be naturally identified as subsets of norm spaces, because all metric spaces can be made a subset of a norm spaces, albeit maybe not all `naturally'.
Jacob DensonJacob Denson
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