A note on the orthogonality equation with two functions

The aim of this paper is to describe the solution (f, g) of the equation ⟨f(x)|g(y)⟩=⟨x|y⟩,x,y∈D,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle{f(x)}|{g(y)}\rangle=\langle{x}|{y}\rangle,\quad x,y\in D,$$\end{document}where f,g:D→Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f,g\colon D\to Y}$$\end{document}, X, Y are Hilbert spaces over the same field K∈{R,C}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}}$$\end{document}, D is a dense subspace of X.