J. Orihuela

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.

A sequential property of set-valued maps

Let X and Y be Hausdorff topological spaces and F: X+ Y a set-valued map. A problem of continuing interest in analysis has been to study under what conditions, for a given point x E X, we can replace Fx by another set Cx in order to have upper semicontinuity in x together with some compactness property in Cx. For instance, if we are dealing with selection problems for F it is useful to have some compact subset Cx with Cx c Fx at a first stage, see [ 121. If we are dealing with extension problems for the range space F(X), it would be useful to have some compact subset Cx with Fx c Cx, see [3]. On the other hand, if we are looking for extension problems in the domain space Xc S, it would be useful to have some compact subset C’s, for s E & with F(N n X) n Cs # 0 for any neighbourhood N of s in S, see [17]. It is surprising that all these problems that have been studied by many different people have a common underlying structure. Our main objective in this paper is to reveal this common structure, and to study conditions to ensure the compactness of the sets Cx in each case. In Section 2 we deal with decreasing sequences of sets and we study the compactness of their sets of cluster points. For this purpose we use some “sequential cluster sets” that have been previously used by Hansell, Jayne, Labuda, and Rogers [12], and by the authors in [3]. In Section 3 we apply these results to obtain the theorems on boundaries of upper semicontinuous set-valued maps stated in [12, 163. In Section 4 we apply these techniques to the problem of extending the range space of a set-valued map obtaining an upper semicontinuous compact set-valued map. See our previous paper [3].

Metrizability of precompact subsets in ( LF )-spaces

In this paper we prove that every precompact subset in any ( LF )-space has a metrizable completion. As a consequence every ( LF )-space is angelic and in this way the answer to a question posed by K. Floret [3] is given. Some contributions to the general problem of regularity in inductive limits posed by K. Floret [3] are also given. Particularly, extensions of well-known results of H. Neuss and M. Valdivia are provided in the general setting of ( LF )-spaces. It should also be noted that our results hold for inductive limits of an increasing sequence of metrizable spaces.

A generic factorization theorem

Let F : Z → X be a minimal usco map from the Baire space Z into the compact space X . Then a complete metric space P and a minimal usco G : P → X can be constructed so that for every dense G δ -subset P 1 of P there exist a dense G δ Z 1 of Z and a (single-valued) continuous map f : Z 1 → P 1 such that F ( Z )⊂ G ( f ( z )) for every z ∈Z 1 . In particular, if G is single valued on a dense G δ -subset of P , then F is also single-valued on a dense G δ -subset of its domain. The above theorem remains valid if Z is Cech complete space and X is an arbitrary completely regular space. These factorization theorems show that some generalizations of a theorem of Namioka concerning generic single-valuedness and generic continuity of mappings defined in more general spaces can be derived from similar results for mappings with complete metric domains. The theorems can be used also as a tool to establish that certain topological spaces contain dense completely metrizable subspaces.

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 (∗).

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.

A Biased View of Topology as a Tool in Functional Analysis

A survey about “Topology as a tool in functional analysis” would be such a giant enterprise that we have, naturally, chosen to give here “Our biased views of topology as a tool in functional analysis”. The consequence of this is that a big portion of this long paper deals with topics that we have been actively working on during the past years. These topics range from metrizability of compact spaces (and their consequences in functional analysis), networks in topological spaces (and their consequences in renorming theory of Banach spaces), distances to spaces of functions (and their applications to the study of pointwise and weak compactness), James’ weak compactness theorem (and their applications to variational problems and risk measures). Some of the results collected here are a few years old while many others are brand new. A few of them are first published here and most of them have been often used in different areas since their publication. The survey is completed with a section devoted to references to some of what we consider the last major achievements in the area in recent years.

Compactness, Optimality, and Risk

This is a survey about one of the most important achievements in optimization in Banach space theory, namely, James’ weak compactness theorem, its relatives, and its applications. We present here a good number of topics related to James’ weak compactness theorem and try to keep the technicalities needed as simple as possible: Simons’ inequality is our preferred tool. Besides the expected applications to measures of weak noncompactness, compactness with respect to boundaries, size of sets of norm-attaining functionals, etc., we also exhibit other very recent developments in the area. In particular we deal with functions and their level sets to study a new Simons’ inequality on unbounded sets that appear as the epigraph of some fixed function f. Applications to variational problems for f and to risk measures associated with its Fenchel conjugate f ∗ are studied.

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.