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Writing A Map As A Composition of Three Linear/Bilinear Maps

  1. Feb 5, 2009 #1
    1. The problem statement, all variables and given/known data
    Consider the map F: C0([a,b],Rn) --> R, F([tex]\phi[/tex])=[tex]\int\phi(t)\phi(t)[/tex]dt (This is the integral from a to b). Write F as a composition of three maps, each of which is linear or bilinear. Then use the chain rule and generalized product rule to compute DF([tex]\phi[/tex]).

    2. Relevant equations

    3. The attempt at a solution
    The only maps I could think of would be something like this:

    [tex]\phi[/tex]: [a,b] --> Rn

    A: C0([a,b],Rn) --> C0([a,b],Rn)

    And then maybe I would need a map for [tex]\phi[/tex][tex]\phi[/tex]? But I'm not sure what that would be or if that's even correct. I'm not even sure that I am on the right track at all with the maps. Does anyone have any ideas?

    Is it possible that F is one of the three maps? So that the composition becomes F of A of phi?
  2. jcsd
  3. Feb 5, 2009 #2


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    Of course, [itex]\int f(t)dt[/itex] is a linear function, from the set of integrable functions to the set of real numbers, itself. Are you including that? And f(x,y)= xy is multi-linear.
  4. Feb 5, 2009 #3
    I'm not sure I know what you mean.:confused:

    Isn't F already that map of integrable functions? Or will I need a more general one such as I: C0([a,b],Rn) --> Rn as one of my maps?

    Then, what I'm thinking is that it should be:

    I of 'something' of phi. I can't figure out what that middle map would be though.
    Last edited: Feb 5, 2009
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