= Weighted inner product
{title2=$\langle f,g\rangle_w$}
For a positive weight $w$, the weighted inner product of <functions> is $\langle f,g\rangle_w=\int\overline f g w$. It is the usual <inner product> in the <L2 space> for the measure $w(x)dx$. A <Sturm-Liouville eigenfunction expansion> uses this weight to make its <eigenfunctions> <orthogonal>.
The weight specifies how different locations contribute to a squared error: $\|f-g\|_w^2=\int|f-g|^2w$. More weight makes an error in that region more costly. The <weighted Euclidean inner product> makes the same choice for a finite list of coordinates.
For an explicit real-function example on $[0,1]$, let $w(x)=1+x$ and approximate $f(x)=x$ by a <constant function>. The <orthogonal projection onto a finite-dimensional subspace> spanned by the function $1$ gives
$$
c=\frac{\langle x,1\rangle_w}{\langle1,1\rangle_w}
=\frac{\int_0^1x(1+x)\,dx}{\int_0^1(1+x)\,dx}
=\frac{5/6}{3/2}=\frac59.
$$
The best unweighted constant is $1/2$. Giving greater weight to larger $x$ moves the best constant upward, so a weighting choice changes the approximation even though the allowed family of <functions> is unchanged.
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