## 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 (∑∞n=1 Xn)E is asymptotically midpoint uniformly convex (AMUC) if

and only if the spaces Xn are uniformly AMUC and E is uniformly monotone.

We also show that Lp(X) is AMUC if and only if X is uniformly convex.

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 (∑∞n=1 Xn)E is asymptotically midpoint uniformly convex (AMUC) if

and only if the spaces Xn are uniformly AMUC and E is uniformly monotone.

We also show that Lp(X) is AMUC if and only if X is uniformly convex.

Original language | English (US) |
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Pages (from-to) | 439-446 |

Number of pages | 8 |

Journal | Bulletin of the Belgian Mathematical Society - Simon Stevin |

Volume | 24 |

Issue number | 3 |

State | Published - 2017 |