Let V denote the vector space of all continuous functions :
f:[0, 1] -> R
Where the 1st and 2nd derivatives exist and are continuous. Suppose further that:
f(0)=f(1)=0
Now let L: V -> V be defined by
L(f) = f''
Define the inner product <f, g> as follows:
<f, g> =
Prove that <L(f), g> = <f, L(g)>
(I've already proved that L is a linear map. The hint on this part says to use integration by parts twice, but I'm not seeing it. Help? If I can get some guidance on the left-hand side, I can do the right hand side as well since it will be similar. Thanks.)