suppose f:[a,b]-> real numbers is continuos and for all continuous functions g:[a,b] -> real numbers, then intergrate from b to a of (f.g)=0. show that f is identically 0.
how do i go about solving this?
Just as a remark, this is true more generally in the sense that if for every (for some arbitrary but fixed ) with then . The proof is not much different, merely take to conclude. I mention this because it shows that separates points with respect to the usual weak topology on it. In this context this theorem (as simple as it is) actually has a name. It's the Fundamental Lemma of the Calculus of Variations.