Show that if $f\in L^{3/2}([0,1])$ satisfies $ \int_0^1x^nf(x)dx = 0,\:\: n\in\mathbb{N} \cup \{0\}$ then $f = 0$ a.e.
I have not taken a course in integration theory, but my guess is that this apply that $ \int_0^1p(x)f(x)dx \approx \int_0^1|f(x)|^2dx =0 $ and hence we get $f(x) = 0$ a.e.
Im a bit unsure about the $"\approx"$ part? How could I write that more rigorously and does it matter that we are in $L^{3/2}([0,1])$ or could we as well be in any $L^p([0,1])$ space?