Let $p,q, \in \mathbb{R}[x]$ Define a inner product space by $$\left \langle p(x),g(x) \right \rangle=\int_{0}^{1}p(x)q(x)dx$$
Show that $W=\{ p(x)\in R[x]: p(0)=0\}$ is a subspace such that $W^\perp=\{0 \}$.
I tried the following:
Let $q \in W^\perp$ so that $\left \langle x,q(x) \right \rangle=0$, $\left \langle x^2,q(x) \right \rangle=0, \cdots ,\left \langle x^n,q(x)\right \rangle=0$ this proces gives me $n$ system of homogeneous equations and if I prove the determinant of the matrix associate is invertible, then the only solution would be $q=0$.
Is this somewhat correct? Any other hint for a different approach would be helpful, thanks.
