Decomposition of functions between Banach spaces in the orthogonality equation

Let E, F be Banach spaces. In the case that F is reflexive we give a description for the solutions (f, g) of the Banach-orthogonality equation ⟨f(x),g(α)⟩=⟨x,α⟩∀x∈E,∀α∈E∗,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \langle f(x),g(\alpha )\rangle =\langle x,\alpha \rangle \qquad \forall x\in E,\forall \alpha \in E^*, \end{aligned}$$\end{document}where f:E→F,g:E∗→F∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:E\rightarrow F,g:E^*\rightarrow F^*$$\end{document} are two maps. Our result generalizes the recent result of Łukasik and Wójcik in the case that E and F are Hilbert spaces.