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

[azdang]

Homework Statement

Consider the map F: C⁰([a,b],Rⁿ) --> R, F(\phi)=\int\phi(t)\phi(t)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(\phi).

Homework Equations

The Attempt at a Solution

The only maps I could think of would be something like this:

\phi: [a,b] --> Rⁿ

A: C⁰([a,b],Rⁿ) --> C⁰([a,b],Rⁿ)

And then maybe I would need a map for \phi\phi? 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?

 

[HallsofIvy]

Of course, \int f(t)dt 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.

 

[azdang]

I'm not sure I know what you mean.

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.