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Strictly Convex Spaces Via Semi-Inner-Product Space Orthogonality

Ellen Torrance
Proceedings of the American Mathematical Society
Vol. 26, No. 1 (Sep., 1970), pp. 108-110
DOI: 10.2307/2036813
Stable URL: http://www.jstor.org/stable/2036813
Page Count: 3
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Strictly Convex Spaces Via Semi-Inner-Product Space Orthogonality
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Abstract

Let (X, |·|) be a normed space, and let [ ·, · ] be any semi-inner-product on it. We show that (X, |·|) is strictly convex if and only if $\|y + z\| > \|y\|$ whenever [ z, y ] = 0 and z ≠ 0, and if and only if [ Ax, x ] ≠ 0 whenever |I + A| ≤ 1 and Ax ≠ 0. The condition that [ z, y ] = 0 can be replaced by a stronger or weaker condition.

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