We are on E the vector space of continuous and 2 times differentiable functions on [0,1] i.e $C^2[0,1]$ I have the sets $V = f$ such that $f(0)=f(1)=0$ and $W = f$ such that $f=f''$ And the function $<f,g> = \int_0^1[{f(t)g(t)+f'(t)g'(t)}]dt$ I have proven that they are Subspaces and that they are orthogonal by a scalar product defined as above
1) I want to prove that they are orthogonal complementary Subspaces So it remains proving they are supplementary Proving their intersection is empty is easy now my problem is how to prove their sum gives E
2) With that I will be able to find the orthogonal projection of any function on W Now if I'm given another set $G_{a,b} = f$ such that $f(0)=a$ and $f(1)=b$ how can i determine $inf_{f€G_{a,b}}\int_0^1{[{f^2(t)+f'^2(t)}]}dt$ I know it's the same as $(inf_{f€G_{a,b}} {||f||}_{<>})^2$ and that
$inf_{f€G_{a,b}}{||f||}_{<>}$ is $d(0,f)$ which is the projection of 0 on G
But I know nothing about G being a Subspace or it's orthogonal so how to calculate the projection???
Any help will be appreciated thanks!!!
Hint 1: You can explicitly describe all the functions in $W$.
Hint 2: given a function $f \in E$, can you describe how to find a function $g \in W$ such that $f-g \in V$? If so you are done.