Fernando Rambla-Barreno

• Universidad de Cádiz

We prove that if X is a Banach space whose Banach-Mazur distance to a Hilbert space is less than 2412+72+6(19+122), then X has the fixed point property for nonexpansive mappings. Also, a renorming of ℓ2 is constructed in order to show the limitations of the usual techniques.

In 1992, Kiendi, Adamy and Stelzner investigated under which conditions a certain type of function constituted a Lyapunov function for some time-invariant linear system. Six years later, it was obtained that this property holds if and only if the Banach space enjoys the self-extension property. However, the knowledge of these spaces needed to be ex...

The incorporation of mobile applications in educational environments generates a large amount of information resulting from the interaction of students with these applications. The analysis of this information can be of significant importance. The teacher may find it useful, as it can help them to make decisions or to assess the process of teaching...

We introduce two Bishop-Phelps-Bollobas moduli of a Banach space which measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollobas theorem in this space. We show that there is a common upper bound for these moduli for all Banach spaces and we present an example showing that this bound is sharp. We prove the continuity of the...

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.

By means of rough convergence, we introduce two new geometric properties in Banach spaces and relate them to Chebyshev centers and some well-known classical properties, such as Kalton's M property or Garkavi's uniform rotundity in every direction.

By using modulus functions we introduce a new concept of density for sets of natural numbers. Consequently, we obtain a generalization of the notion of statistical convergence which is studied and characterized. As an application, we prove that the ordinary convergence is equivalent to the module statistical conver- gence for every unbounded modulu...

We prove the classical Phillips Lemma in the setting of measures defined on effect algebras of sets. This leads to several Vitali-Hahn-Saks-type results for these measures.

We introduce two Bishop-Phelps-Bollob\'as moduli which measure, for a given Banach space, what is the best possible Bishop-Phelps-Bollob\'as theorem in this space. We show that there is a common upper bound for these moduli for all Banach spaces and we present an example showing that this bound is sharp. We prove the continuity of these moduli and...

By using unbounded modulus functions we introduce a new concept of density for sets of pairs of natural numbers. Consequently, we obtain a generalization of the notion of statistical convergence of double sequences which is studied and characterized. As an application, we prove that `Pringsheim convergence' is equivalent to `module statistical conv...

We characterize uniform rotundity in every direction by means of rough convergence. This characterization still holds if we shift to rough statistical convergence. We also study the latter convergence in normed spaces.

We introduce a property of Banach spaces, called uniform convex-transitivity, which falls between almost transitivity and convex transitivity. We will provide examples of uniformly convex-transitive spaces. This property behaves nicely in connection with some vector-valued function spaces. As a consequence, we obtain some new examples of convex-tra...

We study diameter preserving linear bijections from ${\cal C}(X, V)$ onto ${\cal C}(Y, {\cal C}_0(L))$ where $X, Y$ are compact Hausdorff spaces, $L$ is a locally compact Hausdorff space and $V$ is a Banach space. In the case when $X$ and $Y$ are infinite and ${\cal C}_0(L)^*$ has the Bade property we prove that there is a diameter preserving linea...

In this paper we obtain a new version of the Orlicz-Pettis theorem by using statistical convergence. To obtain this result we prove a theorem of uniform convergence on matrices related to the statistical convergence.

We prove that for every member X in the class of real or complex JB∗-triples or preduals of JBW∗-triples, the following assertions are equivalent:(1)X has the fixed point property.(2)X has the super fixed point property.(3)X has normal structure.(4)X has uniform normal structure.(5)The Banach space of X is reflexive. As a consequence, a real or com...

This paper has been withdrawn, since it contains a corollary with impossible consequences and the source of an error is currently unknown.

Let L, S and D denote, respectively, the set of ℚ-linear functions, the set of everywhere surjective functions and the set of dense-graph functions on ℝ. In this note, we show that the sets D∖(S∪L), S∖L, S∩L and D∩L∖S are lineable. Moreover, all these sets contain (omitting zero) a vector space of the biggest possible dimension 2 c .

We study diameter preserving linear bijections from C(X, V) onto C(Y, Z) where X, Y are compact Hausdorff spaces and V, Z are Banach spaces. For instance, we obtain that if X has at least four points, Z is linearly isometric to V and either Z is a C-0(L) space or Z* is strictly convex or smooth, then there is a diameter preserving linear bijection...

We prove that if the one-point compactification of a locally compact, noncompact Hausdorff space L is the topological space called pseudoarc, then C0(L,C) is almost transitive. We also obtain two necessary conditions on a metrizable locally compact Hausdorff space L for C0(L) being almost transitive.

By means of $M$-structure and dimension theory, we generalize some known results and obtain some new ones on almost transitivity in $\mathcal{C}_0(L,X)$. For instance, if $X$ has the strong Banach–Stone property, then almost transitivity of $\mathcal{C}_0(L,X)$ is divided into two weaker properties, one of them depending only on topological propert...

The aim of this paper is to study the set IXr of isometric reflection vectors of a real Banach space X. We deal with geometry of isometric reflection vectors and parallelogram identity vectors, and we prove that a real Banach space is a Hilbert space if the set of parallelogram identity vectors has nonempty interior. It is also shown that every rea...