function in $L_2$ space that is $f=0$ almost everywhere

Question

I have to prove this

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ $\in L^2[0,1]$ and Lebesgue integral $\int_{[0, 1]}f(x)x^n=0$ for all $n=0, 1, 2, ...$ Then $f=0$ almost everywhere.

I think I have learned that by Stone–Weierstrass theorem polynomial is dense in $C[a,b]$, hence polynomials can be used to approximate the function in $L_p$ space. But I'm not sure if this is useful for this problem.

Answer 1

This is indeed useful. Polynomials are dense in $L^2[0,1]$.

Suppose $f_n \to f$ in $L^2$ where $f_n$ is a polynomial.

Then $\int f f_n \mathrm dx = 0$ for all $n$ by your assumption.

By continuity of the $L^2$-scalar product this implies $\int |f|^2 \mathrm dx = 0$, thus $f=0$ almost everywhere.