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
@article{PIM_1984_N_S_35_49_a23,
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/}
}
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/