Consider the space of real polynomials on $[0,1]$ with scalar product $\langle f,g\rangle:=\int_0^1 f(x)g(x)dx$. Let $\{f_1,\dots,f_n\}$ be a set of pairwise orthogonal polynomials.
Assume that $f$ and $g$ satisfy:
$\langle f,g\rangle=0$,
$\deg f=\deg g$,
$f(t)=g(t)$ for some $t\in[0,1]$,
$span\{f_1,\dots,f_n,f\}=span\{f_1,\dots,f_n,g\}$.
Does it imply $f=g$ ?