Show that $f(x)\equiv 0$ if $ \int_0^1x^nf(x)\,dx=0$ Let $f \in C[0,1]$. If for each integer $n \ge 0$ we have 
$$ \int_0^1x^nf(x)\,dx=0$$ show that $f(x) \equiv 0$

The equality implies that $f$ is orthogonal to every polynomial in $L_2[0,1]$, so in particular it is orthogonal to the Legendre polynomials (which form a basis in $L_2[0,1]$). This implies that $f$ must be $0$ a.e., and Since $f$ is continuous, we must conclude that $f \equiv0$.
