C. Angosto

Technical University of Cartagena | UPCT · Departamento de Matemática Aplicada y Estadística

Given a topological space X , we establish formulas to compute the distance from a function f∈RX to the spaces of upper semicontinuous functions and lower semicontinuous functions. For this, we introduce an index of upper semioscillation and lower semioscillation. We also establish new formulas about distances to some subspaces of continuous functi...

We extend the result of B. Cascales et al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit sphere is the union of all its finite-dimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contr...

We extend the result of B. Cascales at al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit ball is the union of all its finite-dimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contrac...

We obtain formulae to calculate the asymptotic center and radius of bounded sequences in ${\cal C}_0(L)$ spaces. We also study the existence of continuous selectors for the asymptotic center map in general Banach spaces. In Hilbert spaces, even a H\"older-type estimation is given.

In recent years, several quantitative counterparts for several classical such as Krein- Smulyan, Eberlein- Smulyan, Grothendieck, etc. have been proved by different authors. These new versions strengthen the original theorems and lead to new problems and applications in topology and analysis. In this survey, we present several of these quantitative...

Let E be a non-Archimedean Banach space over a non-Archimedean locally compact nontrivially valued field K := (K,vertical bar.vertical bar). Let E '' be its bidual and M a bounded set in E. We say that M is epsilon-weakly relatively compact if (M) over bar (sigma(E ''+E')) subset of E+B-E '',B-epsilon where E+B-E '',B-epsilon is the closed ball in...

Oriented interval exchange transformations (i.e.t.s) admitting infinite orbit of the Rauzy–Veech operator are minimal. In this paper, we analyse what happens for non-oriented i.e.t.s. In particular, we prove an analogous result for i.e.t.s with flips (Theorem A). We also show that the behaviour of i.e.t.s with flips is different to the one of orien...

Let E be a Fréchet space, i.e. a metrizable and complete locally convex space (lcs), E" its strong second dual with a defining sequence of seminorms ||.||n induced by a decreasing basis of absolutely convex neighbourhoods of zero Un, and let H ⊂ E be a bounded set. Let ck(H):=sup{d(clustE"(φ),E):φεHN} be the "worst" distance of the set of weak *-cl...

For a Banach space E and its bidual space E ′′, the following function \({k(H) : = {\rm sup}_{y\in\overline{H}^{\sigma(E^{\prime \prime},E^{\prime})}} {\rm inf}_{x\in E} \|y - x\|}\) defined on bounded subsets H of E measures how far H is from being σ(E, E′)-relatively compact in E. This concept, introduced independently by Granero [10] and Cascale...

Given a complete probability space and a Banach space X we establish formulas to compute the distance from a function to the spaces of strongly measurable functions and Bochner integrable functions. We study the relationship between these distances and use them to prove some quantitative counterparts of Pettis’ measurability theorem. We also give s...

Given a metric space X and a Banach space (E, ||·||) we use an index of σ-fragmentability for maps f Î EX{f \in E^X} to estimate the distance of f to the space B 1(X, E) of Baire one functions from X into (E, ||·||). When X is Polish we use our estimations for these distances to give a quantitative version of the well known Rosenthal’s result stati...

Measures of weak noncompactness are formulae that quantify different characterizations of weak compactness in Banach spaces: we deal here with De Blasi's measure ω and the measure of double limits γ inspired by Grothendieck's characterization of weak compactness. Moreover for bounded sets H of a Banach space E we consider the worst distance k(H) of...

We establish here some inequalities between distances of pointwise bounded subsets H of RX to the space of real-valued continuous functions C(X) that allow us to examine the quantitative difference between (pointwise) countable compactness and compactness of H relative to C(X). We prove, amongst other things, that if X is a countably K-determined s...

Many classical results about compactness in functional analysis can be derived from suitable inequalities involving distances to spaces of continuous or Baire one functions: this approach gives an extra insight to the classical results as well as triggers a number of open questions in dierent exciting research branches. We exhibit here, for instanc...

Given a metric space X and a Banach space (E,‖⋅‖) we study distances from the set of selectors Sel(F) of a set-valued map F:X→P(E) to the space B1(X,E) of Baire one functions from X into E. For this we introduce the d-τ-semioscillation of a set-valued map with values in a topological space (Y,τ) also endowed with a metric d. Being more precise we o...