Is is possible to talk about a "space of differential operators"? If one defined such a space would it be possible to talk about limits? I don't really have much background so I'm really just looking for a simple/intuitive explanation?
For example if there was a space of differential operators, and it was normed could I then find limits? If you were to take the sum of two differential operators would you get a new differential operator in that space? I'm sure all these questions have answers I just don't know where to find a good introduction to some of these ideas.
The short answer is, yes, there are spaces of differential operators. You do have to be careful about domains, but in the "nice" cases your operators will all have a common core of smooth functions. These are generally not normed spaces, because the operators are unbounded, but there are concepts of limits. You might start by looking at Reed and Simon, "Methods of Modern Mathematical Physics I: Functional Analysis" http://books.google.ca/books?id=fXX0j4qa8G8C