I came across a definition of Schwartz Space where they were defined as functions $f$ such that $\mathrm{lim}_{|x|\to \infty} |x^{\alpha}D^{\beta}f(x)|=0$ for any pair of multiindices $\alpha,\ \beta$. Now this clearly implies that $P(x)Lf(x)$ is bounded for any polynomial $P$ and differential operator with constant coefficients $L$. How do I prove the converse? If $P(x)Lf(x)$ is bounded for any polynomial $P$ and differential operator with constant coefficients $L$, would it still hold that $\mathrm{lim}_{|x|\to \infty} |x^{\alpha}D^{\beta}f(x)|=0$ for any pair of multiindices $\alpha,\ \beta$?
If not, then $\forall N>0, \exists x_N $ and $ M_N>0, $ so that $ |x_N|>N$ and $|x^{\alpha}D^{\beta}f(x)|>M_N$. How does this contradict boundedness of the same? If $M_N$ were not to depend on $N$, and were fixed, we could contradict boundedness, but as it is I can't see a way out.