Self-adjoint operators on real Banach spaces
Suppose that \((X,\|\cdot\|)\) is a real normed space.The normed derivatives \({\rho}_+^{'},{\rho}_-^{'} :X\times X \to \mathbb{R}\) are defined by \[ {\rho}_{\pm}^{'}(x,y):=\lim_{t\to 0^{\pm}} \frac{\|x+ty\|^2-\|x\|^2}{2t}=\|x\|.\lim_{t\to 0^{\pm}} \frac{\|x+ty\|-\|x\|}{t}. \] In this paper the author shows that if \(X\) is a Banach space and \(f:X\to X\) is a function such that \({\rho}_{+}^{'}(f(x),y)={\rho}_{+}^{'}(x,f(y)),\) then \(f\) is linear and continuous. See page 177 of [\textit{C. Alsina, J. Sikorska} and \textit{M. S. Tomás}, Norm derivatives and characterizations of inner product spaces. Hackensack, NJ: World Scientific. (2010; Zbl 1196.46001)]. He also solves the question,when \(X=C(M)\) and derives a new characterizations of inner product spaces.
characterization of inner product spaces
scientific article; zbMATH DE number 6140763