KURATOWSKI’S INDEX OF NON-COMPACTNESS AND RENORMING IN BANACH SPACES.

F. GARCÍA, L. ONCINA, J. ORIHUELA AND S. TROYANSKI

A point x ∈ A ⊂ (X, ∥ · ∥) is quasi-denting if for every ε > 0 there exists a slice of A containing x with Kuratowski index less than ε, where the Kuratowski index of non-compactness of B is defined by α(B) := inf{ε > 0 : B is covered by a finite family of sets of diameter less thanε}.

Theorem. Let X be a normed space and let F be a norming subspace of its dual. Then X admits an equivalent σ(X, F)-lower semi-continuous LUR norm if, and only if, for every ε > 0 we can write X = ∪_{n=1}^{∞}X_{n,ε} in such a way that for every x ∈ X_{n,ε}, there exists a σ(X, F)-open half space H containing x with α (H ∩ X_{n,ε}) < ε.

Taking advantage of homogeneity one can replace the space X in the former theorem by its unit sphere.

¿From the topological point of view we shall see that a normed space X admits an equivalent σ(X, F)-lower semi-continuous LUR norm if, and only if, the norm topology has a network N = ∪_{n=1}^{∞}N_{n} such that for every n ∈ ℕ and for every x ∈ ∪N_{n} there is a σ(X, F)-open half space H with x ∈ H such that H meets only a finite number of elements from N_{n}, therefore turning the ‘discrete’ condition appearing in [A. Moltó, J. Orihuela, S. Troyanski, and M. Valdivia, On weakly locally uniformly rotund Banach spaces, J. Funct. Anal. 163 (1999), no. 2, 252–271.] into a ‘locally finite’ one.