Strictly convex spaces via semi-inner-product space orthogonality

Torrance, Ellen

doi:10.1090/S0002-9939-1970-0261328-2

Strictly convex spaces via semi-inner-product space orthogonality

Author: Ellen Torrance
Journal: Proc. Amer. Math. Soc. 26 (1970), 108-110
MSC: Primary 46.10
DOI: https://doi.org/10.1090/S0002-9939-1970-0261328-2
MathSciNet review: 0261328
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Abstract: Let $(X,|| \cdot ||)$ be a normed space, and let $[ \cdot , \cdot ]$ be any semi-inner-product on it. We show that $(X,|| \cdot ||)$ is strictly convex if and only if $||y + z|| > ||y||$ whenever $[z,y] = 0$ and $z \ne 0$, and if and only if $[Ax,x] \ne 0$ whenever $||I + A|| \leqq 1$ and $Ax \ne 0$. The condition that $[z,y] = 0$ can be replaced by a stronger or weaker condition.

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