B. Cascales

Weight of precompact subsets and tightness

Pfister (1976) and Cascales and Orihuela (1986) proved that precompact sets in (DF)- and (LM)-spaces have countable weight, i.e., are metrizable. Improvements by Valdivia (1982), Cascales and Orihuela (1987), and Kakol and Saxon (preprint) have varying methods of proof. For these and other improvements a refined method of upper semi-continuous compact-valued maps applied to uniform spaces will suffice. At the same time, this method allows us to dramatically improve Kaplanskys theorem, that the weak topology of metrizable spaces has countable tightness, extending it to include all (LM)-spaces and all quasi-barrelled (DF)-spaces, both in the weak and original topologies. One key is showing that for a large class G including all (DF)- and (LM)-spaces, countable tightness of the weak topology of E in G is equivalent to realcompactness of the weak∗ topology of the dual of E.

The Lindelof property and fragmentability

Let K be a compact Hausdorff space and C(K) the space of continuous real functions on K. In this paper we prove that any tp(K)-Lindelbf subset of C(K) which is compact for the topology tp(D) of pointwise convergence on a dense subset D C K is norm fragmented; i.e., each non-empty subset of it contains a non-empty tp(D)-relatively open subset of small supremum norm diameter. Several applications are given.

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.

Compactoid filters and USCO maps

The aim of this paper is to report in a short and self-contained way on the properties of compactoid and countably compactoid filters. We apply them to some questions in both topology and analysis such as the generation and extension of USCO maps, the study of some properties of K-analytic spaces and the study of bounds for the weight of compact sets in spaces obtained through inductive operations.

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.

Measurability and semi-continuity of multifunctions

The following pages contain details of a mini-course of three lectures given at the V International Course of Mathematical Analysis of Andalucia (CIDAMA), Almeria, September 12-17, 2011. When I was invited to give this mini-course and thought about possible topics for it, I decided to talk about multifunctions because they have always been present in my research on fields theoretically apart from each other as topology and integration theory. Therefore you will find here my biased views regarding part of the research that I have done over the years. The proofs for this material have been published elsewhere by me or by some other authors. This mini-survey is written attending to the invitation of the publishers of this book with the sole purpose of witnessing the given mini-course and with the aim of providing the reader with connections and ideas that usually are not written in research papers. I thank the organizers of CIDAMA V as well as the editors of the book for their kind invitation to give the lecture and write this mini-survey. In these notes we shall deal with multifunctions (or set-valued maps). Multifunctions naturally appear in analysis and topology, for instance via inequalities, performing unions or intersections with sets indexed in another set, considering the set of points minimizing an expression, etc. First, we will present some results about semi-continuity of multifunctions, namely, lower semi-continuity and an application of Michael’s selection theorem. Then we will deal with upper semi-continuity of multifunctions and an application to the generation of K-analytic structures with consequences in topology and functional analysis. We will finish by showing a few results about measurability for multifunctions related to the Kuratowski-Ryll-Narzesdky selection theorem and their implications to integrability of multifunctions for non separable Banach spaces.