Abstract
The aim of this paper is to give an answer to a question posed by Alsina, Sikorska and Tomás. Namely, we show that, under suitable assumptions, a function \(f:X\!\rightarrow \!Y\) from a normed space X into a normed space Y, satisfying the functional equation
has to be a linear similarity (scalar multiple of a linear isometry).
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References
Alsina, C., Sikorska, J., Tomás, M.S.: Norm Derivatives and Characterizations of Inner Product Spaces. World Scientific, Hackensack (2010)
Amir, D.: Characterization of Inner Product Spaces. Birkhäuser Verlag, Basel (1986)
Chmieliński, J., Wójcik, P.: On a \(\rho \)-orthogonality. Aequ. Math. 80, 45–55 (2010)
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Wójcik, P. On a functional equation characterizing linear similarities. Aequat. Math. 93, 557–561 (2019). https://doi.org/10.1007/s00010-018-0603-2
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DOI: https://doi.org/10.1007/s00010-018-0603-2
Keywords
- Functional equation
- Normed spaces
- Norm derivatives
- Smoothness
- Orthogonality in normed spaces
- Height function