On asymptotic structure the Szlenk index and UKK properties in Banach spaces. Positivity 3
| Author(s) : | Th. Schlumprecht E. Odell H. Knaust, |
| Publisher : | N/A |
| Publication Date : | 1999 |
| ISSN : | N/A |
| Abstract : | Abstract. Let B be a separable Banach space and let X = B ? be separable. We prove that if B has finite Szlenk index (for all >0) then B can be renormed to have the weak * uniform Kadec-Klee property. Thus if >0 there exists ()> 0sothatif(xn) is a sequence in the ball of X converging ? to x so that lim infn? ? ?xn ? x ? ? then ?x ? ? 1 ? (). In addition we show that the norm can be chosen so that () ? c p for some p< ? and c>0., |