Rotund complex normed linear spaces

Beesack, P. R.; Hughes, E.; Ortel, M.

doi:10.1090/S0002-9939-1979-0529209-2

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Rotund complex normed linear spaces
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by P. R. Beesack, E. Hughes and M. Ortel PDF

Proc. Amer. Math. Soc. 75 (1979), 42-44 Request permission

Abstract:

We show that rotundity in a complex normed linear space is equivalent to the property that for any distinct vectors x and y of unit norm, a complex number $\alpha$ may be found for which $\left \| {\alpha x + (1 - \alpha )y} \right \| < 1$. This leads to a natural proof of a result due to Taylor and Foguel on the uniqueness of Hahn-Banach extensions.

References

A. E. Taylor, The extension of linear functionals, Duke Math. J. 5 (1939), 538–547. MR 345
Shaul R. Foguel, On a theorem by A. E. Taylor, Proc. Amer. Math. Soc. 9 (1958), 325. MR 93696, DOI 10.1090/S0002-9939-1958-0093696-3