Homework Help: Derivative w.r.t. the Lp norm

1. Aug 13, 2009

foxjwill

1. The problem statement, all variables and given/known data
Let $$A:L^2(0,\infty)\to L^2(0,\infty)$$ be given by $$f(x)\mapsto f(x+1)$$. What is the derivative $$A'$$, if it exists, of $$A$$? That is, we want a function $$A':L^2(0,\infty)\to L^2(0,\infty)$$ such that
$$\lim_{\|h\|\to0}\frac{\left\|A(f+h)-Af -hA'f\right\|}{\|h\|}=0.$$​

2. Relevant equations

3. The attempt at a solution
I was able to simplify the differentiability condition thingy to
$$\lim_{\|h\|\to0}\frac{\left\|h(x+1) -h(x)A'f\right\|}{\|h(x)|}=0.$$​
However, I'm not sure where to go from there. Any hints?