TITLE: Vector space for which finite-dimensionality is open
QUESTION [7 upvotes]: Is there a vector space for which it is unknown whether it is finite-dimensional or infinite-dimensional?
This vector space should be somehow "naturally occuring" in the sense that it is interesting as a vector space itself. Otherwise one could take e.g. the set of all twin primes (or any set of which finiteness is unknown) and consider it as a free vector space over some field to get a trivial example.

REPLY [7 votes]: I don't know if it's a satisfying answer, but if you take a real number $\alpha$, its transcendence on $\mathbb{Q}$ is equivalent to $\mathbb{Q}[\alpha]$ being of infinite dimension. So for instance, the dimension of $\mathbb{Q}[\gamma]$ is unknown.
While this seems like a crafted example (and it is), looking at $\mathbb{Q}[\alpha]$ is absolutely natural when studying the algebraic properties of a real number.