Let $C^1_0[a,b]:=\{f \ C^1[a,b]|f(a)=f(b)=0\}.$ Providing $C^1_0[a,b]$ is dense in $L^2[a,b]$, I want to prove the following statement:
if for $g,h\in L^2[a,b]$,
$\int_a^b g \phi \,dx =\int_a^b h \phi \,dx$ for all test functions $\phi\in C^1_0[a,b]$,
then $g = h$ almost everywhere.
It seems to be similar with fundamental lemma of calculus of variation, how can I extend the the result to $L^2$ functions? My guess is using density argument, along with one of convergence theorems, but I failed to construct the proof.
Please help me out here. Thank you.