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The norm induced by the specified inner product is the L2 normA polynomial $p_n\in {\Pi }_n$ is a best polynomial least-squares approximation whenMinimizing this norm is equivalent to minimizing its square, the integrated squared error. Continuity ensures that a function with zero squared norm is identically zero: a nonzero value would give a neighbourhood on which the square has a positive integral.

A polynomial $p_n\in {\Pi }_n$ is a best polynomial least-squares approximation when it minimizes the weighted error norm over all polynomials of degree at most $n$:Equivalently it minimizes ${\int }_0^1{x}^{-1/2}(f(x)-q(x){)}^{2}dx$. The weight is integrable, so the expression is finite for continuous functions; it defines a genuine norm because a continuous function with zero weighted squared integral must be identically zero.

A polynomial least-squares approximation $p_n\in {\Pi }_n$ minimizes the weighted residual norm:Equivalently it minimizes ${\int }_{-1}^1{x}^2(u(x)-q(x){)}^{2}dx$. This is a best approximation within a finite-dimensional polynomial space, without an interpolation requirement.

Define the reflected polynomial $\tilde{p}(x)=-p(-x)$. It belongs to ${\Pi }_n$. Since $u$ is odd, a change of variable and the evenness of the weight giveThus $\tilde{p}$ is another polynomial least-squares approximation. By uniqueness, $\tilde{p}=p$, soThe conclusion includes the zero polynomial, which is both odd and even.