I am wondering the link as the title implies. The Spring 87 problem in Berkeley Problems in Mathematics is as follows:
Let $V$ be a finite dimensional linear subspace of $C^{\infty}(\mathbb{R})$. Assume that $V$ is closed under differentiation. Prove that there is a constant coefficient operator $L=\sum^{n}_{k=0}a_{k}D^{k}$ such that $V=\{f:Lf=0\}$.
I am wondering why the finite dimensional condition given would imply such a strong result. Because if we assume the required relationship the reverse is not necessarily true(the whole space is obviously closed under any differential operator), I feel something deeper may be buried in this problem I do not know. A hint or an illuminating example would be mostly welcome. I just do not know how to attack this problem.