Suppose we are given a subset $A$ of a complex vector space $\mathcal V$, and we are asked to look at the subspace $V$ (Hamel-) spanned by $A$. Of course, a spanning set may be very large compared to the dimension of its span, so my question is: what sort of conditions guarantee that $V$ is finite-dimensional?
Let's say that $A$ isn't something we can muck around with, but that we may make any niceness assumptions on the ambient space that we wish. So, one easy thing we can do is
- if $\mathcal V$ is finite-dimensional, then $V$ is always finite-dimensional.
Something more interesting occurs in normed spaces:
- if $\mathcal V$ is normed, then $V$ is finite-dimensional if its closed unit ball is compact.
(I tried to find a list with Google, and I'd be kind of surprised if nothing like this exists, but I wasn't able to get the search words to find it. So a reference to some compiled list would also be nice.)
Context: I am working on a homework question for a Lie theory course. The details of the question don't seem particularly important here, but we construct an action on $C^\infty(G,\Bbb C)$ for each element of $G$, take linear combinations, and are asked to show that this set is finite-dimensional.