Stanimir Troyanski

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.

Renormings of C(K) spaces

We survey the current state of research in renormings of C(K) spaces.ResumenEn este artículo analizamos el estado actual de la investigaciíon en teoría del renormamiento de espacios C(K).

Strictly convex norms and topology

We introduce a new topological property called (∗) and the corresponding class of topological spaces, which includes spaces with Gδ-diagonals and Gruenhage spaces. Using (∗), we characterize those Banach spaces which admit equivalent strictly convex norms, and give an internal topological characterization of those scattered compact spaces K, for which the dual Banach space C(K) ∗ admits an equivalent strictly convex dual norm. We establish some relationships between (∗) and other topological concepts, and the position of several well-known examples in this context. For instance, we show that C(K) ∗ admits an equivalent strictly convex dual norm, where K is Kunen’s compact S-space. Also, under additional axioms, we provide examples of compact scattered non-Gruenhage spaces of cardinality ℵ1 having (∗).

Polyhedrality in pieces

The aim of this paper is to present two tools, Theorems 4 and 7, that make the task of finding equivalent polyhedral norms on certain Banach spaces easier and more transparent. The hypotheses of both tools are based on countable decompositions, either of the unit sphere SX or of certain subsets of the dual ball BX⁎ of a given Banach space X. The sufficient conditions of Theorem 4 are shown to be necessary in the separable case. Using Theorem 7, we can unify two known results regarding the polyhedral renorming of certain C(K) spaces, and spaces having an (uncountable) unconditional basis. New examples of spaces having equivalent polyhedral norms are given in the final section.

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.

LUR renormings through Deville's Master Lemma

A completely geometrical approach for the construction of locally uniformly rotund norms and the associated networks on a normed space X is presented. A new proof providing a quantitative estimate for a central theorem by M. Raja, A. Moltó and the authors is given with the only external use of Deville-Godefory-Zizler decomposition method.ResumenPresentamos una aproximación completamente geométrica para la construcción de normas localmente uniformemente convexas y sus network asociadas en un espacio normado X. Se da una nueva demostración, con estimaciones cuantitativas, de un resultado central de M. Raja, A. Moltó y los autores usando únicamente el método de descomposición de Deville-Godefroy-Zizler.

Moduli of convexity and smoothness of reflexive subspaces of L1

Abstract We show that for any probability measure μ there exists an equivalent norm on the space L 1 ( μ ) whose restriction to each reflexive subspace is uniformly smooth and uniformly convex, with modulus of convexity of power type 2. This renorming provides also an estimate for the corresponding modulus of smoothness of such subspaces.

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