Least Squares Approximation for odd functions question

Question

Suppose that $f \in C[-1,1]$ is an odd function on $[-1,1]$. Show that polynomial $p_n$ of least squares approximation for $f$ in the norm $|\cdot|$ is an odd function on $[-1,1]$.

Answer 1

If the(!) least square approximation for $f(x)$ is $p_n(x)$, then the least square approximation for $-f(-x)$ is $-p_n(-x)$.