PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 35(49), pp. 173177 (1984) 

ON CHARACTERIZATIONS OF INNERPRODUCT SPACESO. P. Kapoor and Jagadish PrasadDepartment of Mathematics, I.I.T. Kanpur, IndiaAbstract: The generalized innerproduct $(x,y)$ in a normed linear space $X$ is the right Gateaux derivative of the functional $\x\^2/2$ at $x$ in the direction of $y$. The orthogonality relation for the generalized innerproduct is $x\perp_G y\Leftrightarrow (x,y)=0$. Tapia has proved that $X$ must be an innerproduct space if the generalized innerproduct is either symmetric or linear in $y$, and Detlef Laugwitz showed that if dimension $X\geq 3$ and the orthogonality for generalized innerproduct is symmetric, then $X$ is an innerproduct space. In this note we discuss this orthogonality relation and provide alternative proofs of the results of Tapia and Laugwitz. Classification (MSC2000): 46B99 Full text of the article:
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