Infinite dimensional spaces other than functional spaces

Question

"Functional analysis" is the study of infinite dimensional spaces equipped with inner product, norm, topology...etc. The most interesting spaces are the spaces of functions/operators and sequences.

I don't know if there's "another kind" of infinite dimensional spaces other than space of functions/operators/sequences which is interesting.

Answer 1

$\mathbb{R}$ is an infinite dimensional vector spaces over $\mathbb{Q}$, but it is not an Hilbert space. You can see that all the axioms for a vector space are verified if you define the sum of two ''vectors" as the usual sum of real numbers and the product for a scalar $q \in \mathbb{Q}$ as the usual product.

This space has an infinite dimensional Hamel basis.

And, obviously, any $\mathbb{R}^n$ is finite dimensional over $\mathbb{R}$ but infinite dimensional over $\mathbb{Q}$.

Answer 2

In fact all normed spaces are subspaces of some function spaces. This could be the reason why functional analysis has its name.

Answer 3

One example might be $M(K)$, the space of all regular Borel measures on $K$ of finite variation, where $K$ is compact space. This space arises as the dual of $C(K)$.