In topology, there are of course many different infinite-dimensional topological vector spaces over $\mathbb R$ or $\mathbb C$. Luckily, in algebraic topology, one rarely needs to worry too much about the distinctions between them. Our favorite one is:
- $\mathbb R^\infty = \cup_{n < \omega} \mathbb R^n$, the "smallest possible" infinite-dimensional space
Occasionally one might also use:
$\mathbb R^\omega = \prod_{n < \omega} \mathbb R$
$\ell_2(\mathbb R)$
If one is doing serious geometry, using differential equations or something, one might start to use more interesting TVS in a serious way, more like an analyst.
If things get really crazy, one might see "uncountable-dimensional" analogs of these, but it's pretty rare for this to come up in my (probably quite skewed!) experience.
Typically, an algebraic topologist needs such a space either to receive an embedding from a smaller space, or to collect together all finite-dimensional vector spaces in one place. When fancier vector spaces come up, it's usually because one is interfacing with other fields, and one ends up actively looking for ways to simplify them.
Question 1: Just how misleading was that synopsis of the use of infinite-dimensional vector spaces in algebraic topology?
One thing that algebraic geometry shares with algebraic topology is an emphasis on fairly "finite objects" -- this is probably even more the case in algebraic geometry since even mapping spaces can have relevant finiteness properties. So my guess would be that infinite-dimensional vector spaces are likely used in algebraic geometry in a similar way to how they're used in algeraic topology.
Question 2: Just how off-target is that guess -- that algebraic geometry uses infinite-dimensional vector spaces in the "same" way as algebraic topology?
More specifically, in algebraic geometry, there are natural candidates for analogs to the first two spaces above:
$\mathbb A^\infty_0 = \cup_n \mathbb A^n$ (colimit taken as a scheme, if it exists)
$\mathbb A^\infty = \cup_n \mathbb A^n$ (an ind-scheme)
$\operatorname{Spec} k[x_0,x_1,\dots] = \mathbb A^\omega = \prod_{n<\omega} \mathbb A^1$ (as pointed out by Wojowu in the comments, this product is preserved by the inclusion of affine schemes into schemes.)
I would guess that these schemes are non-isomorphic (probably $\mathbb A^\infty$ is not compact, and in particular not affine?) as in topology, but that the difference between them is typically inessential for "most" algebro-geometric concerns, as I think it is in algebraic topology.
Question 3: Is that correct? Are $\mathbb A^\infty$, $\operatorname{Spec} k[x_0,x_1,\dots]$, and $\mathbb A^\omega$ non-isomorphic, but with differences which are inessential "in practice"?
Question 4: Are there other infinite-dimensional vector spaces which come up in algebraic geometry?
I also suspect that in algebraic geometry, some of the roles played by $\mathbb R^\infty$ in algebraic topology are really more appropriately analogized to the roles of $\mathbb P^\infty$, e.g. to receive embeddings from varieties. I'm not sure how to fit that issue into the above broad statements, but it probably means that I should be asking about $\mathbb P^\infty = \cup_n \mathbb P^n$ and $\operatorname{Proj} k[x_0,x_1,\dots]$ too.
