Consider the linear space of continuous functions $C[-1,+1]$ defined over the interval $[-1,+1]$. We define an inner product $\langle\cdot , \cdot\rangle$ on $C[-1,+1]$ by
$$\langle f,g\rangle=\int_{-1}^{+1} f(x)\cdot g(x) \, d(x),$$
for any $f,g$ in $C[-1,+1]$.
1) Consider the linear sub-space $V=\operatorname{Span}(x, x^2)$ in $C[-1,+1]$. Find an orthonormal basis of $V$.
2) Consider the projection $\operatorname{Proj}_V:\,C[-1,+1]\rightarrow V$. Use the orthonormal basis obtained in 1) to calculate $\operatorname{Proj}_V(x^3)$.
I get that $\langle x,x^2\rangle =\int_{-1}^{+1} x^3\,dx=0$, but I'm not sure what the dimension of an orthonormal basis would be. $2\times 2$?