In this article that I am reading, I am given a non-negative spectral function $w(\lambda)$ which is "interpreted as a weight function determining the scalar product of two functions $f(\lambda)$ and $g(\lambda)$" where $$(f,g) := \int_0^\infty w(\lambda)f(\lambda)g(\lambda)\;\;d\lambda.$$
Here $f(\lambda)$ and $g(\lambda)$ are functions of the class $L_w^2(0,\infty)$. If all moments of the spectral function exist then we can construct a set of orthogonal polynomials $p_n(\lambda)$ such that $$(p_m,p_n) = h_n\delta_{mn}$$ where $\delta_{nm}$ is the Kronecker delta and $h_n$ is the $L_w^2(0,\infty)$ norm of $p_n(\lambda)$.
I have not seen this norm notation before. What is the significance of the $w$ subscript? How would you write this norm out as a summation?
Apologies if this question has been asked but I have looked and can't seem to find this notation.