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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,\vert\vert \cdot \vert\vert)$ be a normed space, and let $ [ \cdot , \cdot ]$ be any semi-inner-product on it. We show that $ (X,\vert\vert \cdot \vert\vert)$ is strictly convex if and only if $ \vert\vert y + z\vert\vert > \vert\vert y\vert\vert$ whenever $ [z,y] = 0$ and $ z \ne 0$, and if and only if $ [Ax,x] \ne 0$ whenever $ \vert\vert I + A\vert\vert \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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0261328-2
Keywords: Strictly convex space, semi-inner-product space, orthogonality
Article copyright: © Copyright 1970 American Mathematical Society

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