Geodesic
On Characterizations of Inner-product Spaces
Publications de l'Institut Mathématique, _N_S_35 (1984) no. 49, p. 173 .

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The generalized inner-product $(x,y)$ 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 inner-product is $x\perp_G y\Leftrightarrow (x,y)=0$. Tapia has proved that $X$ must be an inner-product space if the generalized inner-product is either symmetric or linear in $y$, and Detlef Laugwitz showed that if dimension $X\geq 3$ and the orthogonality for generalized inner-product is symmetric, then $X$ is an inner-product space. In this note we discuss this orthogonality relation and provide alternative proofs of the results of Tapia and Laugwitz.
Classification : 46B99 
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     author = {O. P. Kapoor and Jagadish Prasad},
     title = {On {Characterizations} of {Inner-product} {Spaces}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {173 },
     publisher = {mathdoc},
     volume = {_N_S_35},
     number = {49},
     year = {1984},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1984_N_S_35_49_a23/}
}
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O. P. Kapoor; Jagadish Prasad. On Characterizations of Inner-product Spaces. Publications de l'Institut Mathématique, _N_S_35 (1984) no. 49, p. 173 . http://geodesic.mathdoc.fr/item/PIM_1984_N_S_35_49_a23/