Here is my attempt at the first few lines...can someone just let me know if it is correct or not? (It's probably not lol)
(D(v1+v2))(r)=(v1+v2)'(r)
=(v1(r)+v2(r))'
=(v1(r))'+(v2(r))'
Define D ∈ L(p(R), p(R)) by (Dp)(r)=p'(r) for all r∈R and for all p∈p(R).
I know that to show that something is a linear transformation you need to show that F(v1+v2)=F(v1)+F(v2) and also F(av)=aF(v). However I do not understand HOW to show this. Can someone help with this please? I just am not sure how exactly to get started. It would be great if perhaps someone could do the first few lines or so, I think I would be able to finish if I could only get a start.