In my Analysis class we keep using the symbol $D$ to stand for differentiation in our analysis course, but what is $D$ itself, really?
- nLab seems to say its a functor, but for some reason requires smoothness (as oposed to e.g. $C^1$ or even just differentiability).
- Dieudonné, in his Treatise on Analysis, Vol. 4, p. 127, talks about using Lie Groups to generalize "the operators of differentiation".
- Bourbaki, in their Lie Groups and Lie Algebras, Ch. 3 §17, have this definition:
- This paper uses "the differentiation operator" in the context of "weighted spaces of holomorphic functions".
- This other paper says "the differentiation operator $D$ is defined by $Df=f'$, [...]"
- There are some related questions on here too:
- Wikipedia talks about differential operators generalizing the differentiation operator. It also has this article about generalizing the derivative.
- Of course, many authors just introduce $D$ as a notation and then move on without mentioning it again.
I literally just want to know what $D$ is so that I can continue with my analysis lectures without feeling like I'm already using things without knowing what they are. I tried asking something similar before (see here), but I suspect the question was unclear.
In the very possible case that there's different approaches to what $D$ might be (i.e. different, incompatible generalizations), some clarity would still be much appreciated. Thank you!