Let X and Y be normed spaces over F∈{R,C}\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {F}} \in \{{\mathbb {R}}, {\mathbb {C}}\}$$\end{document} and f:X→Y\documentclass[12pt]{minimal}
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\begin{document}$$f :X \rightarrow Y$$\end{document} a surjective mapping. Suppose that |ϕf(y)(f(x))|=|ϕy(x)|\documentclass[12pt]{minimal}
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\begin{document}$$|\phi _{f(y)}(f(x))|=|\phi _y(x)|$$\end{document} holds for all x,y∈X\documentclass[12pt]{minimal}
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\begin{document}$$x,y\in X$$\end{document} and all support functionals ϕf(y)\documentclass[12pt]{minimal}
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\begin{document}$$\phi _{f(y)}$$\end{document} at f(y) and ϕy\documentclass[12pt]{minimal}
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\begin{document}$$\phi _y$$\end{document} at y, or equivalently, suppose that for all semi-inner products on X and Y, compatible with given norms, |[f(x),f(y)]|=|[x,y]|\documentclass[12pt]{minimal}
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\begin{document}$$\vert [f(x), f(y)] \vert = \vert [x, y] \vert $$\end{document} holds for all x,y∈X\documentclass[12pt]{minimal}
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\begin{document}$$x,y \in X$$\end{document}. Then f=σU\documentclass[12pt]{minimal}
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\begin{document}$$f=\sigma U$$\end{document}, where σ:X→F\documentclass[12pt]{minimal}
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\begin{document}$$\sigma :X \rightarrow {\mathbb {F}}$$\end{document} is a phase function, and U:X→Y\documentclass[12pt]{minimal}
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\begin{document}$$U :X \rightarrow Y$$\end{document} is a linear or a conjugate linear isometry.
