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 $\displaystyle u_i , v_i \in C[a,b]$