Let $X$ and $P$ be linear operators on a $\mathbb C$ vector space $V$ and $I$ the identity operator. Suppose the commutator
$$[X,P] = XP - PX = ciI$$
for some real, positive constant $c$.
What can we say about the dimension of $V$?
The motivation for this problem was a throw away line by a theoretical physicist in a lecture I saw where he said this necessarily implied $V$ is infinite dimensional. $X$ and $P$ will be recognized by anyone with training in quantum mechanics as labels for two popular operators. However their particularities shouldn't matter as I want to see what we can say about $V$ with just this information.
Any ideas?