Weighted inner product ($⟨f,g{⟩}_w$)

For a positive weight $w$, the weighted inner product of functions is $⟨f,g{⟩}_w=\int \overline{f}gw$. 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.