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What can we say about $f$ if $\int_0^1 f(x)p(x)dx=0$ for all polynomials $p$?
I want to prove that for $f \in L^1(\mathbb{T})$ s.t $\forall g \in C(\mathbb{T}) \ \int_{\mathbb{T}} fg = 0$ then $\int_{\mathbb{T}} |f| = 0$.
(where $\mathbb{T}$ is the unit circle).
How to show this?
I thought approximating $f$ by some $h$ continuous, i.e $||f-h||_1 \leq \epsilon$, but I don't see how to procceed from here, any hints or full solutions will be appreciated.
Thanks.