On Mathematics Stack Exchange, I asked the following question: [Why are infinite-dimensional vector spaces usually equipped with additional structure?](https://math.stackexchange.com/questions/4751895/why-are-infinite-dimensional-vector-spaces-usually-equipped-with-additional-stru) 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.)



Why do infinite-dimensional vector spaces usually have additional structure?
Here is a supplement to the nice [answer](https://math.stackexchange.com/a/4751952/87579) 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](https://mathoverflow.net/a/452860), 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.


