Numerical index and renorming

The Szlenk Index and the Fixed Point Property under Renorming

Assume that X is a Banach space such that its Szlenk index Sz X is less than or equal to the first infinite ordinal ω. We prove that X can be renormed in such a way that X with the resultant norm satisfies R X < 2, where R · is the Garcı́a-Falset coefficient. This leads us to prove that if X is a Banach space which can be continuously embedded in a Banach space Y with Sz Y ≤ ω, then, X can be re...

Renorming James Tree Space

We show that James tree space JT can be renormed to be Lipschitz separated. It negatively answers the question of J. Borwein, J. Giles and J. Vanderwerff whether every Lipschitz separated Banach space is an Asplund space.

Trees in Renorming Theory

Renorming spaces with greedy bases

In approximation theory one is often faced with the following problem. We start with a signal, i.e., a vector x in some Banach space X. We then consider the (unique) expansion ∑∞ i=1 xiei of x with respect to some (Schauder) basis (ei) of X. For example, this may be a Fourier expansion of x, or it may be a wavelet expansion in Lp. We then wish to approximate x by considering m-term approximatio...

A note on convex renorming and fragmentability

Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b) If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c) If the Banach space is weakly ...