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$\phi$-orthogonally additive mappings. I. (English) Zbl 0763.46022
One says that the elements $x,y\in X$ of a vector space $X$ over an Euclidean ordered field are $\varphi$-orthogonal if $\varphi(x,y)=0$ for a non-isotropic bilinear form $\varphi$. In the paper this concept is treated and is generalized for vector spaces over arbitrary fields, sesquilinear forms with respect to the automorphisms of the field and for orthogonally additive mappings with values in Abelian groups.

##### MSC:
 46C50 Generalizations of inner products
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##### References:
 [1] M. Fochi, Functional equations onA-orthogonal vectors,Aequationes Math.,38 (1989), 28--40. · Zbl 0679.39006 · doi:10.1007/BF01839491 [2] A. Pinsker, Sur une fonctionnelle dans l’espace de Hilbert,C. R. Acad. Sci. URSS N. S.,20 (1938), 411--414. · Zbl 0020.37001 [3] J. Rätz, On orthogonally additive mappings,Aequationes Math.,28 (1985), 35--49. · Zbl 0569.39006 · doi:10.1007/BF02189390 [4] J. Rätz, On orthogonally additive mappings, II,Publicationes Math. Debrecen.,35 (1988), 241--249. [5] J. Rätz and Gy. Szabó, On orthogonally additive mappings, IV,Aequationes Math.,38 (1989), 73--85. · Zbl 0679.39005 · doi:10.1007/BF01839496 [6] W. Scharlau,Quadratic and Hermitian Forms, Springer (Berlin-Heidelberg-New York-Tokyo, 1985). · Zbl 0584.10010 [7] K. Sundaresan and O. P. Kapoor,T-orthogonality and nonlinear functional on topological vector spaces,Can. J. Math.,25 (1973), 1121--1131. · Zbl 0268.46012 · doi:10.4153/CJM-1973-120-7 [8] F. Vajzović, On a functional which is additive onA-orthogonal pairs,Glasnik Mat.,21 (1966), 75--81.