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.
Citation
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
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