Total differential of general function mapping

1. Apr 13, 2012

birulami

I am looking for an explanation and derivation of a total differential of a 2nd order function, i.e. a function that maps one function to another.

To be more specific, lets say I have a function $l:ℝ^n\to ℝ$ that I use to define a 2nd order function $L:(ℝ^k\to ℝ^n) \to (ℝ^k\to ℝ)$ as $L(f) := l\circ f$ for every $f:ℝ^k\to ℝ^n$.

Given that I have sufficiently useful norms defined on a function space $ℝ^i\to ℝ^j$, I assume that the total differential, $D(L,f)$ is well defined and is, for each "point" $f:ℝ^k\to ℝ^n$ of the function space a linear function (map) with the same algebraic type as $L$, i.e. $D(L,f):(ℝ^k\to ℝ^n) \to (ℝ^k\to ℝ)$.

My hunch is that $D(L,f)$ can be expressed in terms of the total differentials of $l$ and $f$, if they are uniformly continuous, but I just cannot derive the solution.

Can someone confirm, that the total differential of $L$ is a well defined concept?
What is the solution in terms of $l$ and $f$?