A special version of the Schwarz lemma on an infinite dimensional domain (Q1375125)

Let \((E,||)\) be a complex Banach space, \(B:=\{x\in E\: |x|<1\}\). Assume that any point from \(\partial B\) is a complex extreme point for \(\overline B\). Let \(f\:B\longrightarrow B\) be a holomorphic mapping such that \(f(0)=0\) and \(|f(w)|=|w|\) for \(w\in U\), where \(U\) is a nonempty open subset of \(B\). Then \(f\) is a linear isometry. The case \(E=\mathbb C^{n}\) was proved by \textit{J.-P. Vigué} [Indiana Univ. Math. J. 40, No. 1, 293-304 (1991; Zbl 0733.32025)]. The proof of the general case follows Vigué's methods.