An Asymptotic Property of Schachermayer's Space under Renorming

✍ Scribed by Denka Kutzarova; Denny H. Leung

• Book ID: 102591988
• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 118 KB
• Volume: 250
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2000.7104

✦ Synopsis

Let X be a Banach space with closed unit ball B. Given k ∈ , X is said to be k-β, respectively, k + 1 -nearly uniformly convex ( k + 1 -NUC), if for every ε > 0 there exists δ, 0 < δ < 1, so that for every x ∈ B and every ε-separated sequence x n ⊆ B there are indices

It is shown that a Banach space constructed by Schachermayer is 2-β, but is not isomorphic to any 2-NUC Banach space. Modifying this example, we also show that there is a 2-NUC Banach space which cannot be equivalently renormed to be 1-β.