Sums of asymptotically midpoint uniformly convex spaces. (English) Zbl 1388.46018

Dilworth, S. J.; Kutzarova, Denka; Randrianarivony, N. Lovasoa; Romney, Matthew

Bull. Belg. Math. Soc. - Simon Stevin 24, No. 3, 439-446 (2017).

Summary: 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.

MSC:

46B20

Geometry and structure of normed linear spaces

Keywords:

uniform convexity; asymptotic geometry; asymptotic moduli; AMUC