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 $\langle f(x),g(α)\rangle=\langle x,α\rangle\hspace{10mm}\forall x\in E,\forall α\in E^*,$ where $f:E\rightarrow F,g:E^*\rightarrow F^*$ are two maps. Our result generalizes the recent result of Łukasik and Wójcik in the case that $E$ and $F$ are Hilbert spaces.

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