Antonio J. Guirao

Open problems in the geometry and analysis of banach spaces

Preface.- Basic linear structure.- Basic linear geometry.- Biorthogonal systems.- Smoothness, smooth approximation.- Nonlinear geometry.- Some more nonseparable problems.- Some applications.- Bibliography.- List of concepts and problems.- Symbol index.- Subject index.

On Preserved and Unpreserved Extreme Points

An extreme point of the closed unit ball of a Banach space is said to be preserved if it is extreme of the closed unit ball of the bidual space; otherwise it is called unpreserved. The beginning of the present work takes the form of a survey on this topic, presenting some elementary facta about those concepts—usually with new proofs—and discussing in particular Katznelson’s solution to a Phelps’ question on preserved extreme points, not available, to our knowledge, in the literature. In a second part, some new results are presented. Since some of them depend on the concept of polyhedrality, we first review several results on this topic. Then we present Godun renorming theorem for the class of nonreflexive Banach spaces, and Morris renorming result—with a new proof—on separable Banach spaces containing a copy of (c_0). We show that, under some extra conditions—polyhedrality—a similar renorming, this time adding smoothness, can be defined ensuring strict convexity with all points in the unit sphere unpreserved extreme. We finalize this work by presenting what—to our knowledge—is the first nonseparable result of this kind for the natural class of the weakly compactly generated Banach spaces.

Some More Nonseparable Problems

The following is the definition of a type of Schauder basis that works also for nonseparable spaces. It is due to P. Enflo and H. P. Rosenthal in [EnRo73].

Differentiability and Structure, Renormings

In this chapter we review some problems on smoothness, rotundity, and its connection to the structure of spaces. We recommend, for example, [BenLin00, DeGoZi93, Fa97, FHHMZ11, HMVZ08], and the recent book [HaJo14] for this area.

Basic Linear Geometry

A subset K of a Banach space X is said to be a Chebyshev set if every point in X has a unique nearest point in K. In such a case, the mapping that to x ∈ X associates the point in K at minimum distance is called the metric projection.

Basic Linear Structure

A sequence {e i }i = 1 ∞ in a Banach space X is called a Schauder basis for X if for each x ∈ X there is a unique sequence of scalars {α i }i = 1 ∞ such that (x =sum _{ i=1}^{infty }alpha _{i}e_{i}). If the convergence of this series is unconditional for all x ∈ X (i.e., any rearrangement of it converges), we say that the Schauder basis is unconditional . This is equivalent to say that under any permutation (pi: mathbb{N} rightarrow mathbb{N}), the sequence {eπ(i)}i = 1 ∞ is again a basis of X.

Sobre la adaptación, aplicación y evaluación de cursos abiertos masivos online de Matemáticas en la plataforma EdX

En los ultimos anos, con el apoyo de la Universidad Politecnica de Valencia, se han desarrollado diversos MOOCs, con el objetivo de facilitar el aprendizaje y/o ampliar los conocimientos que necesitan los estudiantes. En este marco, con la colaboracion del DMA, hemos presentado en la plataforma UPV[X] una coleccion de cuatro MOOC’s titulados Bases Matematicas: Numeros y Terminologia, Derivadas, Integrales y Algebra que cubren los conocimientos basicos de matematicas que los estudiantes necesitan el primer ano de los diferentes grados en ingenieria. Desde hace pocos meses la UPV es miembro de la red EdX y una de las principales consecuencias ha sido el aumento del alcance de los cursos de Bases Matematicas. Esto ha significado una necesaria adaptacion de los contenidos, una metodologia mucho mas progresiva y una mejora de la forma de evaluacion. En el articulo se presentan las modificaciones realizadas en el proceso de adaptacion a la plataforma EdX, y los resultados actuales comparandolos con los datos obtenidos en anos anteriores y proponiendo mejoras en la evaluacion de los MOOC’s. Se ha elaborado un sondeo con la herramienta correspondientexa0 asi como un analisis de las opiniones de los estudiantes y de los resultados obtenidos por los mismos.