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september 2017 Sums of asymptotically midpoint uniformly convex spaces
S. J. Dilworth, Denka Kutzarova, N. Lovasoa Randrianarivony, Matthew Romney
Bull. Belg. Math. Soc. Simon Stevin 24(3): 439-446 (september 2017). DOI: 10.36045/bbms/1506477692

Abstract

We study the property of asymptotic midpoint uniform convexity for infinite direct sums of Banach spaces, where the norm of the sum is defined by a Banach space $E$ with a 1-unconditional basis. We show that a sum $(\sum_{n=1}^\infty X_n)_E$ is asymptotically midpoint uniformly convex (AMUC) if and only if the spaces $X_n$ are uniformly AMUC and $E$ is uniformly monotone. We also show that $L_p(X)$ is AMUC if and only if $X$ is uniformly convex.

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S. J. Dilworth. Denka Kutzarova. N. Lovasoa Randrianarivony. Matthew Romney. "Sums of asymptotically midpoint uniformly convex spaces." Bull. Belg. Math. Soc. Simon Stevin 24 (3) 439 - 446, september 2017. https://doi.org/10.36045/bbms/1506477692

Information

Published: september 2017
First available in Project Euclid: 27 September 2017

zbMATH: 06803441
MathSciNet: MR3706812
Digital Object Identifier: 10.36045/bbms/1506477692

Subjects:
Primary: 46B20

Keywords: AMUC , asymptotic geometry , asymptotic moduli , uniform convexity‎

Rights: Copyright © 2017 The Belgian Mathematical Society

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Vol.24 • No. 3 • september 2017
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