Legendre polynomials in $L^2[-1,1]$

Question

I came across a problem that asked to show there's no proper subset of Legendre polynomials in $L^2[-1,1]$ that is complete. Since by Weierstrass Approximation Theorem, Legendre polynomials are complete in $L^2[-1,1]$, so my attempt so far is to show that the Legendre polynomials actually form a basis of $L^2[-1,1]$, then there won't be a proper subset. Am I thinking right? Any help or hint is appreciated.

Answer 1

Suppose you have an orthonormal set $\{ e_n \}_{n=1}^{\infty}$ in a Hilbert space $\mathcal{H}$. If you remove $e_1$ from the set, then the remaining set cannot be complete because $e_1$ is orthogonal to the remaining set.