Does f.g means f multiplied by g?
You can pick any g, so let g = f
Now what does the statement simplify to?
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.