I'm looking at up at how to properly define a differential operator, that is, explicitly write it as a mapping between function spaces.
Well, Wikipedia defines it by setting a mapping $A$ between function spaces $\mathcal{F}_1 ,\mathcal{F}_2$ and. It goes on to say that a given a $u\in\mathcal F_1$, a $k$-order PDO finitely generated by $u$ is defined as: \begin{equation} P(x,\partial_x)=\sum_{|\alpha|\leq k}a_{\alpha}(x)\ \partial^{\alpha}_x \end{equation}
For what I figure to be some $a\in\mathcal F_1$. Whereas the equation itself feels natural, I'm having trouble understanding the purpouse of the preliminaries on the definition. That is, what's the need to define a preliminar map that serves the purpose of some function of it's domain generating the operator. I suppose what I'm asking is: Why do we need to construct something to generate the operator instead of simply defining: \begin{equation} P:\mathcal{F}_1\rightarrow\mathcal{F}_2 \end{equation}
I'd guess that it has to do with the multi-index, but I can't formalize it. Maybe, my question will reduce to just what it properly means that a function "generates" a diff operator.
Any help is appreciated.