= Weighted norm
{title2=$\|f\|_w^2=\int |f(x)|^2w(x)\,dx$}
A positive measurable weight $w$ defines the displayed <norm> on the <L2 space> for the <measure> $w(x)\,dx$, with functions identified when equal almost everywhere. This is the <norm> induced by the <weighted inner product>. Changing $w$ changes the contribution of each region to approximation error. If $p_0,\ldots,p_N$ are nonzero mutually <orthogonal> real functions, completing squares shows that the unique best approximation to real $f$ in their <linear span> has coefficients $a_j=\langle f,p_j\rangle_w/\|p_j\|_w^2$.
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