1. ## Bilinear Forms

Here's the question:

Part (a)

I think to show bilinearity, I must prove these two conditions (left linear form & right linear form):

$\displaystyle \int^{b}_{a} f(\lambda u_1+ \mu u_2,v)g(\lambda u_1+ \mu u_2,v)$

$\displaystyle \int^{b}_{a} f(u, \lambda v_1 + \mu v_2)g(u, \lambda v_1 + \mu v_2)$

Can anyone show me how we can prove this, so I can get started? Thanks.

Edit: $\displaystyle \forall \lambda, \mu \in IR$ and $\displaystyle u_i , v_i \in C[a,b]$

2. I don't understand why the functions $\displaystyle f,g$ in your post are functions of two variables. You want to show that the operator $\displaystyle I$ is a bilinear form, not the functions $\displaystyle f,g$.

You want to show that

1) $\displaystyle I(f+g,h)=I(f,h)+I(g,h)$

2) $\displaystyle I(f,h+g)=I(f,h)+I(f,g)$

3) $\displaystyle I(af,g)=I(f,ag)=aI(f,g)$ for any $\displaystyle a \in \mathbb{R}$.

3. Originally Posted by Bruno J.
I don't understand why the functions $\displaystyle f,g$ in your post are functions of two variables. You want to show that the operator $\displaystyle I$ is a bilinear form, not the functions $\displaystyle f,g$.

You want to show that

1) $\displaystyle I(f+g,h)=I(f,h)+I(g,h)$

2) $\displaystyle I(f,h+g)=I(f,h)+I(f,g)$

3) $\displaystyle I(af,g)=I(f,ag)=aI(f,g)$ for any $\displaystyle a \in \mathbb{R}$.
Ah! I see what you mean! But where did the "h" come form?

I think this is what I need to prove:

$\displaystyle I(\lambda f_1 + \mu f_2, g) = \lambda I (f_1 , g) + \mu I (f_2 , g)$

$\displaystyle I(f, \lambda g_1 + \mu g_2) = \lambda I (f, g_1) + \mu I (f , g_2)$

Could you give me any clues on how to prove these?