On Uniformly Convex and Uniformly Kadec-Klee Renomings
Summary: We give a new construction of uniformly convex norms with a power type modulus on super-reflexive spaces based on the notion of dentability index. Furthermore, we prove that if the Szlenk index of a Banach space is less than or equal to \(\omega\) (first infinite ordinal) then there is an equivalent weak\(^*\) lower semicontinuous positively homogeneous functional on \(X^*\) satisfying the uniform Kadec-Klee property for the weak\(^*\)-topology (UKK\(^*\)). Then we solve the UKK or UKK\(^*\) renorming problems for \(L^p(X)\) spaces and \({\mathcal C}(K)\) spaces for \(K\) a scattered compact space.
- Institute of Mathematics and Informatics Bulgaria
- Bulgarian Academy of Sciences Bulgaria
Isomorphic theory (including renorming) of Banach spaces, Scattered Compact, dentability index, scattered compact space, super- reflexive spaces, \(L^ p\) spaces, uniformly convex norms, weak\(^*\)-topology, Lp Spaces, Szlenk Index, Dentability, Renorming, Kadec-Klee, Szlenk index, Geometry and structure of normed linear spaces, power type modulus, Kadec-Klee property, Uniformly Convex, Super-Reflexive
Isomorphic theory (including renorming) of Banach spaces, Scattered Compact, dentability index, scattered compact space, super- reflexive spaces, \(L^ p\) spaces, uniformly convex norms, weak\(^*\)-topology, Lp Spaces, Szlenk Index, Dentability, Renorming, Kadec-Klee, Szlenk index, Geometry and structure of normed linear spaces, power type modulus, Kadec-Klee property, Uniformly Convex, Super-Reflexive
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