Let $U$ be an open set of the Riemann sphere, $z_i$ be $n$ distinct points of $U$, and $E$ the vector space of meromorphic functions on $U$ with poles of order no more than 2.
Let $F$ be the subspace of $E$ whose elements are holomorphic in a neighborhood of the $z_i$.
Does $E/F$ have finite dimension ? If so, what is it ?
It is clear that it has dimension at least $2n$, since the $\frac{1}{(z-z_i)^k}$, $k=1,2$, form a free family. However, I couldn't determine if there was more (intuition suggests not).