I was reading about functional analysis and it interested me greatly applying the techniques of analysis to function spaces but it seems to have abruptly stopped when getting to the derivative. I know the generalized concept of a derivative on a vector space is called a Frechet derivative.
So what would be the Frechet derivative of a functional operator (a function on standard real functions)? For example what is the Frechet derivative of the basic real analysis derivative? I believe it would just be the the basic derivative since that it already a linear operator. Are there any more more interesting Frechet derivatives of other non-linear functional operators? Is there a name for this?