I have again something from Stein-Shakarchi I would really appreciate some help with. Any references are also welcome!
Suppose $L$ is a linear partial differential operator with constant coefficients. Show that when $d \geq 2$, the linear space of solutions $u$ of $L(u)=0$ with $ u \in C^{\infty}(\mathbb{R}^{d})$ is not finitely dimensional.
Thanks in advance!
EDIT: $L$ takes the form $L= \sum_{|\alpha| \leq n}{a_{\alpha}\left(\frac{\partial{}}{\partial{x}}\right)^{\alpha}}$ with $a_{\alpha} \in \mathbb{C}$ constants.