Orthogonality equation with two unknown functions

The orthogonality equation involving two unknown functions ⟨f(x)|g(y)⟩=⟨x|y⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{ll}\langle{f(x)}|{g(y)\rangle}=\langle{x}|{y}\rangle\end{array}$$\end{document}and, among related problems, the orthogonality preserving property x⊥y⇒f(x)⊥g(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{ll}x\bot y \Rightarrow f(x)\bot g(y)\end{array}$$\end{document}are considered.