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Derivative w.r.t. the Lp norm

  1. Aug 13, 2009 #1
    1. The problem statement, all variables and given/known data
    Let [tex]A:L^2(0,\infty)\to L^2(0,\infty)[/tex] be given by [tex]f(x)\mapsto f(x+1)[/tex]. What is the derivative [tex]A'[/tex], if it exists, of [tex]A[/tex]? That is, we want a function [tex]A':L^2(0,\infty)\to L^2(0,\infty)[/tex] such that
    [tex]\lim_{\|h\|\to0}\frac{\left\|A(f+h)-Af -hA'f\right\|}{\|h\|}=0.[/tex]​

    2. Relevant equations

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