Person: Suárez Granero, Antonio
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First Name
Antonio
Last Name
Suárez Granero
Affiliation
Universidad Complutense de Madrid
Faculty / Institute
Ciencias Matemáticas
Department
Análisis Matemático Matemática Aplicada
Area
Análisis Matemático
Identifiers
9 results
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Now showing 1 - 9 of 9
Item Complementation and embeddings of c0(I) in Banach spaces(Proceedings of the London Mathematical Society, 2002) Argyros, Spiros A.; Castillo, Jesús F.; Suárez Granero, Antonio; Jiménez Sevilla, María Del Mar; Moreno, José PedroItem Convex sets in Banach spaces and a problem of Rolewicz(Studia Mathematica, 1998) Suárez Granero, Antonio; Jiménez Sevilla, María Del Mar; Moreno, José PedroLet BX be the set of all closed, convex and bounded subsets of a Banach space X equipped with the Hausdor metric. In the rst part of this work we study the density character of BX and investigate its connections with the geometry of the space, in particular with a property shared by the spaces of Shelah and Kunen. In the second part we are concerned with the problem of Rolewicz, namely the existence of support sets, for the case of spaces C(K).Item On the Kunen-Shelah properties in Banach spaces(Studia Mathematica, 2003) Suárez Granero, Antonio; Jiménez Sevilla, María Del Mar; Montesinos, Alejandro; Moreno, José Pedro; Plichko, AnatolijWe introduce and study the Kunen-Shelah properties KSi, i = 0, 1,..., 7. Let us highlight for a Banach space X some of our results: (1) X ∗ has a w ∗-nonseparable equivalent dual ball iff X has an ω1-polyhedron (i.e., a bounded family {xi}i<ω1 such that xj / ∈ co({xi: i ∈ ω1 \ {j}}) for every j ∈ ω1) iff X has an uncountable bounded almost biorthonal system (UBABS) of type η, for some η ∈ [0, 1), (i.e., a bounded family {(xα, fα)}1≤α<ω1 ⊂ X × X ∗ such that fα(xα) = 1 and |fα(xβ) | ≤ η, if α = β); (2) if X has an uncountable ω-independent system then X has an UBABS of type η for every η ∈ (0, 1); (3) if X has not the property (C) of Corson, then X has an ω1-polyhedron; (4) X has not an ω1-polyhedron iff X has not a convex right-separated ω1-family (i.e., a bounded family {xi}i<ω1 such that xj / ∈ co({xi: j < i < ω1}) for every j ∈ ω1) iff every w ∗-closed convex subset of X ∗ is w ∗-separable iff every convex subset of X ∗ is w ∗-separable iff µ(X) = 1, µ(X) being the Finet-Godefroy index of X (see [1]).Item On the Nonseparable Subspaces of J(η) and C([1, η])(Mathematische Nachrichten, 2001) Suárez Granero, Antonio; Jiménez Sevilla, María Del Mar; Moreno, José PedroLet η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 2(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c0(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω1) (resp. C([1,ω1])) contains a complemented copy of 2(ω1) (resp. c0(ω1)). As consequence, we give examples (J(ω1), C([1,ω1]), C(V ), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i. e., for every subspace Y ⊆ X we have that Dens (Y) = w∗ –Dens (Y ∗)), in spite of these spaces are not weakly Lindelof determined (WLD).Item Mazur intersection properties and differentiability of convex functions in Banach spaces(Journal of the London Mathematical Society, 2000) Georgiev, P. G.; Suárez Granero, Antonio; Moreno, José Pedro; Jiménez Sevilla, María Del MarIt is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on [script l]1(Γ) and [script l][infty infinity](Γ) are Fréchet differentiable on a dense Gδ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein–Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property.Item Sequential continuity in the ball topology of a Banach space(Indagationes Mathematicae, 1999) Suárez Granero, Antonio; Jiménez Sevilla, María Del Mar; Moreno, José PedroConsider the isometric property (P): the restriction to the unit ball of every bounded linear functional is sequentially continuous in the ball topology. We present in this paper a systematic study of this property, which is a sequential version of the well known ball generated property. A separable Banach space is Asplund if and only if there exists an equivalent norm with property (P). However, in nonseparable spaces this equivalence does not work. Examples of nonseparable Banach spaces with property (P) which are not Asplund are given. Also, we exhibit a nonseparable Asplund space, namely the space of continuous functions on the Kunen compact, admitting no equivalent norm with this property. We characterize reflexive spaces as those satisfying that every equivalent norm has property (P), thus improving a previous characterization involving the ball generated property. Finally, we investigate the relationships between property (P) and Grothendieck spaces.Item On ω-independence and the Kunen-Shelah property(Proceedings of the Edinburgh Mathematical Society, 2002) Suárez Granero, Antonio; Jiménez Sevilla, María Del Mar; Moreno, José PedroWe prove that spaces with an uncountable omega-independent family fail the Kunen-Shelah property. Actually, if {x(i)}(iis an element ofI) is an uncountable w-independent family, there exists an uncountable subset J.C I such that x(j) is not an element of (conv) over bar({x(i)}(iis an element ofj/{j}) for every j is an element of J. This improves a previous result due to Sersouri, namely that every uncountable omega-independent family contains a convex right-separated subfamily.Item Geometry of Banach spaces with property β(Israel Journal of Mathematics, 1999) Suárez Granero, Antonio; Jiménez Sevilla, María Del Mar; Moreno, José PedroWe prove that every Banach space can be isometrically and 1-complementably embedded into a Banach space which satisfies property β and has the same character of density. Then we show that, nevertheless, property β satisfies a hereditary property. We study strong subdifferentiability of norms with property β to characterize separable polyhedral Banach spaces as those admitting a strongly subdifferentiable β norm. In general, a Banach space with such a norm is polyhedral. Finally, we provide examples of non-reflexive spaces whose usual norm fails property β and yet it can be approximated by norms with this property, namely (L 1[0,1], ‖·‖1) and (C(K), ‖·‖∗) whereK is a separable Hausdorff compact spaceItem Intersections of closed balls and geometry of Banach spaces(Extracta Mathematicae, 2004) Suárez Granero, Antonio; Jiménez Sevilla, María Del Mar; Moreno, José PedroIn section 1 we present definitions and basic results concerning the Mazur intersection property (MIP) and some of its related properties as the MIP* . Section 2 is devoted to renorming Banach spaces with MIP and MIP*. Section 3 deals with the connections between MIP, MIP* and differentiability of convex functions. In particular, we will focuss on Asplund and almost Asplund spaces. In Section 4 we discuss the interplay between porosity and MIP. Finally, in section 5 we are concerned with the stability of the (closure of the) sum of convex sets which are intersections of balls and with Mazur spaces.