Aníbal Moltó

A nonlinear transfer technique for renorming

?-Continuous and Co-?-continuous Maps.- Generalized Metric Spaces and Locally Uniformly Rotund Renormings.- ?-Slicely Continuous Maps.- Some Applications.- Some Open Problems.

Kadec and Krein–Milman properties

Abstract The main goal of this paper is to prove that any Banach space X with the Krein–Milman property such that the weak and the norm topology coincide on its unit sphere admits an equivalent norm that is locally uniformly rotund.

On locally uniformly rotund renormings in C(K) spaces

A characterization of the Banach spaces of type C(K) which admit an equivalent locally uniformly rotund norm is obtained, and a method to apply it to concrete spaces is developed. As an application the existence of such renorming is deduced when K is a Namioka{Phelps compact or for some particular class of Rosenthal compacta, results recently obtained in [3] and [6] that were originally proved with methods developed ad hoc.

Finitely fibered Rosenthal compacta and trees

We study some topological properties of trees with the interval topology. In particular, we characterize trees which admit a 2-fibered compactification and we present two examples of trees whose one-point compactifications are Rosenthal compact with certain renorming properties of their spaces of continuous functions.

Topological Properties of the Continuous Function Spaces on Some Ordered Compacta

Some new classes of compacta K are considered for which C(K) endowed with the pointwise topology has a countable cover by sets of small local norm-diameter.

Some Open Problems

We have extensively considered here the use of Stones theorem on the paracompactness of metric spaces in order to build up new techniques to construct an equivalent locally uniformly rotund norm on a given normed space X. The discreetness of the basis for the metric topologies gives us the necessary rigidity condition that appears in all the known cases of existence of such a renorming property [Hay99, MOTV06]. Our approximation process is based on co-σ-continuous maps using that they have separable fibers, see Sect. 2.2. We present now some problems that remain open in this area. Some of them are classical and have been asked by different authors in conferences, papers and books. Others have been presented in schools, workshops, conferences and recent papers on the matter and up to our knowledge they remain open. The rest appear here for the first time. We apologize for any fault assigning authorship to a given question. Rather than to formulate precise evaluation for the first time the problems were proposed, our aim is to provide good questions for young mathematicians entering in the field, we think they deserve all our attention to complete the state of the art in renorming theory.

σ-Continuous and Co-σ-continuous Maps

In this chapter we isolate the topological setting that is suitable for our study. We first present 2.1–2.3 to follow an understandable logical scheme nevertheless the main contribution are presented in 2.4–2.7 and our main tool will be Theorem 2.32. An important concept will be the σ-continuity of a map Φ from a topological space (X, T) into a metric space (Y, g). The σ-continuity property is an extension of continuity suitable to deal with countable decompositions of the domain space X as well as with pointwise cluster points of sequences of functions Φn : X → Y, n = 1,2,… When (X,T) is a subset of a locally convex linear topological space we shall refine our study to deal with σ-slicely continuous maps, the main object of these notes. When (X, T) is a metric space too we shall deal with σ-continuity properties of the inverse map Φ_1 that we have called co-σ-continuity

σ-Slicely Continuous Maps

All examples of σ-slicely continuous maps are connected somehow with LUR Banach spaces. It is clear that if x is a denting point of a set D and Φ is a norm continuous map at x then Φ is slicely continuous at x. Hence if X is a LUR normed space then every norm continuous map Φ on B X is slicely continuous on S X .

Generalized Metric Spaces and Locally Uniformly Rotund Renormings

A class of generalized metric spaces is a class of spaces defined by a property shared by all metric αspaces which is close to metrizability in some sense [Gru84]. The s-spaces are defined by replacing the base by network in the Bing-Nagata-Smirnov metrization theorem; i.e. a topological space is a αspace if it has a αdiscrete network. Here we shall deal with a further re- finement replacing discrete by isolated or slicely isolated. Indeed we will see that the identity map from a subset A of a normed space is A of a normedslicely continuous if, and only if, the weak topology relative to A has a s-slicely isolated network. If A is also a radial set then we have that the identity map Id : (X, weak) ǁ (X,→) is A of a normedslicely continuous and Theorem 1.1 in the Introduction says that this is the case if, and only if, X has an equivalent LUR norm. After our study of this class of maps we can now formulate the following theorem and its corollaries summarizing different characterizations f LUR renormability for a Banach space.