Just what *is* $D$ in Analysis

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:
• 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!
• Letters are not always used in the same way! There are many "usual" ways in which the letter $D$ is used in analysis, many contradicting each other! Thats not a problem, as long as you are making sure that people reading know what you mean by setting up notation carefully. – s.harp Apr 26 at 12:57
• The letters $D$ and $d$ stand for "differential" or "derivation" in these contexts and it is really natural and established to use them when you have something you call a differential. In analysis and geometry I like to use the letter $D$ for the derivative of a map, ie if $f:M\to N$ then $Df: TM\to TN$ is its derivative. The letter $d$ is usually used for the exterior derivative in this context. But its really entirely arbitrary and people do what they want. There is no canonical meaning to "D". – s.harp Apr 26 at 13:05
• @BlondCafé You haven't given us the context of what kind of analysis class you're in, but the simplest possible interpretation is that $D$ is a (linear) function from the vector space of differentiable functions on $\mathbb R$ to the vector space of functions on $\mathbb R$ which admit an antiderivative. In any case, the conclusion you should draw from all these comments along the lines of "impossible to say without more info" is that you should ask your professor. – Kevin Arlin Apr 26 at 19:21