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

1. Feb 5, 2009

### azdang

1. The problem statement, all variables and given/known data
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($$\phi$$).

2. Relevant equations

3. 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?

2. Feb 5, 2009

### HallsofIvy

Staff Emeritus
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.

3. Feb 5, 2009

### 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.

Last edited: Feb 5, 2009