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.)