Why do infinite-dimensional vector spaces usually have additional structure?

Question

On Mathematics Stack Exchange, I asked the following question: Why are infinite-dimensional vector spaces usually equipped with additional structure? Although it received one good answer, I feel that there is more to be said, and a more technical explanation would be welcome. I thus ask a modified version of my question here.

Finite-dimensional vector spaces have a range of applications in pure mathematics. Although infinite-dimensional vector spaces are also widely studied, say, in functional analysis, it seems that most of the time they appear "naturally", they have additional structure, such as an inner product, norm, or a topology. My question is why this phenomenon occurs. Is there a reason for why "pure" infinite-dimensional vector spaces are not more pervasive?

(I also welcome answers that challenge the premise of the question. Perhaps finite-dimensional vector spaces are also most useful in applications when they are equipped with extra structure, or perhaps there are areas of mathematics which do make use of "pure" vector spaces, including infinite-dimensional ones.)

Answer 1

Here is a supplement to the nice answer that you got at MSE.

Much of the theory of infinite dimensional vector spaces is motivated by solving concrete problems in analysis. To solve differential equations, it is often profitable to use vector spaces of functions, and it is for this purpose that the theory of Banach spaces and other areas of functional analysis were developed. It is no surprise that in an analytic context one is concerned with questions of distance/absolute value, defining infinite sums, different notions of convergence etc.

On the other hand, as noted in Mikhail's response, the algebraic theory of infinite dimensional vector spaces is not particularly interesting on its own. For the most part, one of the following two things happens

1. There is an entirely analogous theory to the finite dimensional case (e.g. there is one isomorphism class of vector space for each cardinality of set; a linear transformation is determined uniquely by its values on a basis; a linear transformation is invertible iff the kernel and cokernel vanish etc.).
2. There is no possible analogous theory to the finite dimensional case, without introducing some notion of convergence/topology (e.g. there aren't infinite dimensional determinants that you can use to detect invertibility).

This makes the algebraic theory less interesting than the analytic one.

In summary, the theory of infinite dimensional vector spaces has an analytic flavor because the historical motivations and applications are analytic, and most of the new nontrivial theory lies in an analytic direction.

Edit: One final comment: so-called "pure" infinite dimensional vector spaces actually do appear in mathematical practice quite frequently (at least in algebra). But there typically aren't classes or books devoted specifically to them, for the reasons mentioned above.

Answer 2

We can get some insight into this question by considering matroid theory.

But first, I think the question is phrased in a somewhat misleading way: "Why is there not much interesting theory of infinite-dimensional vector spaces with no extra structure, whereas there is an interesting theory of finite-dimensional vector spaces with no extra structure?" In my opinion, this is not a fair comparison, because finite-dimensionality is the extra structure. In other words, the basic object is a vector space, with no mention of cardinality. It should not be too surprising that if we impose some extra structure on a vector space—namely, finite-dimensionality—then we get a richer theory than if we don't impose any extra structure. All over mathematics, we find that finiteness hypotheses yield rich dividends.

But in the particular case of vector spaces, we can say a little more. Matroids are a generalization of vector spaces, obtained by writing down axioms for (linear) independence. From its beginnings, matroid theory has focused almost exclusively on finite matroids. Given the success of matroid theory, it is natural to ask, why doesn't there seem to be a well-developed theory of infinite matroids? Part of the answer is that one things that makes matroid theory work so well is duality. In the finite case, it is easy to see how to define duality and to derive important consequences from duality. In the infinite case, though, it turns out to be a very difficult problem to develop a satisfactory duality theory. It was not until 2010 that a viable candidate for infinite matroid theory with duality was found, in the paper Axioms for infinite matroids, by Bruhn et al. This struggle in matroid theory perhaps sheds some light on why vector spaces with no extra structure do not have a very rich theory at this time.

Answer 3

An infinite-dimensional vector space without any additional structure carries no more information than the cardinality of its Hamel basis. In modern mathematics, infinite-dimensional vector spaces are studied in a number of fields such as analysis, functional analysis, etc. To go beyond set theory, one needs additional structure. For example, as soon as you add a Hilbert space structure, you get the basic framework for Fourier series, quantum mechanics, etc.

A finite-dimensional vector space is also characterized by the cardinal of its basis, but finite cardinals (i.e., natural numbers) have more structure than infinite ones, such as their semiring structure. This has immediate applications such as topological K-theory, etc.

Answer 4

This remark might perhaps be of interest, despite being tangential to your query. Given how daunting the Schwartz approach to distributions, which involved duality theory for rather exotic locally convex spaces of test functions, was, there were attempts to simplify it by defining distributions to be linear forms on the space of smooth functions of compact support, without demanding continuity. This is tantamount to ignoring the topology on the latter. As I recall, it was feasible to use this approach to obtain a basic theory which was sufficient for the requirements of (some) mathematical physicists. This is perhaps not surprising, given that there are versions of set theory under which all linear functionals on such function spaces are automatically continuous (Solovay, Schwartz, Garnir). Unfortunately, at the moment I have no access to resources which would allow me to provide references but the comprehensive multi-volume treatise on mathematical physics by R. Hermann springs to mind.

Answer 5

In many situations, extra structure is used only as tool to answer questions which have nothing to do with that structure. For example, let $P(D)(f)=\sum_{|\alpha|\le m} c_\alpha \partial^\alpha f$ be linear partial differential operator of finite order with constant coefficients. For every convex subset $\Omega\subseteq \mathbb R^d$, $P(D)$ is a surjective linear map $C^\infty(\Omega)\to C^\infty(\Omega)$.

This theorem of Malgrange has nothing to do with the extra structure of $C^\infty(\Omega)$ being a Fréchet space which is a key ingredient in the proof.

Answer 6

This question is almost similar to the following question. Why do we need more structures on sets, for example vector space or manifold etc? In mathematics people develop notions like sets or infinite dimensional vector spaces not only for the intention for generalisations. These notions should also help us to extract deep mathematical facts. Many times infinite dimensional vector spaces arise from differential equations, operator algebras or abstract algebra etc. Often they have some extra structures and to study them we impose those extra structures. Also often our guiding examples are finite dimensional. For example if you look at local theory of Banach spaces. The idea is to consider finite dimensional phenomenon which are dimension independent. Often we should be able to take limits in infinite dimensional spaces at least for analysts. So we need some more criterion to make us enable to take limits. This idea is prevalent in all analysis where we consider infinite dimensional vector space. Also to talk about limits we need some notion of topology and the space has to be complete also. Thus analysts more or less consider topological vector spaces which are complete.

Answer 7

It seems that people are often taught that mathematics is the study of sets and functions. The truth is that topological spaces are more relevant in practice, and continuous functions are more important than unrestricted functions. The reason is that for some topological spaces $(X, T)$, all computable functions are also continuous. The truth of this statement has been made rigorous for certain topological spaces (Second Countable + Locally Compact?). Speculating more generally, when making the statement rigorous, the topological spaces in question need to be restricted in certain ways, namely to satisfy some separation axiom and countability axiom. The separation axiom is usually sobriety, which is weaker than Hausdorff. The countability axiom might be as weak as Lindelöf. Anyway, this rules out many topologies on vector spaces of infinite dimension.

If one adopts the philosophy that sets+functions are wrong - and topological spaces+continuous functions are right - then this undermines the assumption the question makes that algebraic vector spaces are any more basic than topological vector spaces.