On Characterizations of Inner-product Spaces

O._P. Kapoor, Jagadish Prasad

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.