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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 $f=g$ ?

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    $\begingroup$ If $f = g$ then necessarily $f = g = 0$ because $f \perp g$. $\endgroup$ – mechanodroid Nov 15 '17 at 22:06

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