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Let A be an algebra of continuous real functions on a topological space X. We study when every nonzero algebra homomorphism phi:A --> R is given by evaluation at some point of X. In the case that A is the algebra of rational functions (or real-analytic functions, or C(m)-functions) on a Banach space, we provide a positive answer for a wide class of spaces, including separable spaces and super-reflexive spaces (with nonmeasurable cardinal)., DGICYT, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we study the connections between moduli of asymptotic convexity and smoothness of a Banach space, and the existence of high order differentiable bump functions or equivalent norms on the space. The existence of a high order uniformly differentiable bump function is related to an asymptotically uniformly smooth renorming of power type. On the other hand, the asymptotic uniform convexity of power type is related to the existence of high order smoothness of Nakano sequence spaces., DGES, Bulgarian NFR, Fundación Séneca, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We introduce UDSp-property (resp. UDTq-property) in Banach lattices as the property that every normalized disjoint sequence has a subsequence with an upper p-estimate (resp. lower q-estimate). In the case of rearrangement invariant spaces, the relationships with Boyd indices of the space are studied. Some applications of these properties are given to the high order smoothness of Banach lattices, in the sense of the existence of differentiable bump functions., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We show that any super-reflexive Banach space is a LAMBDA-space (i.e., the weak-polynomial convergence for sequences implies the norm convergence). We introduce the notion Of kappa-space (i.e., a Banach space where the weak-polynomial convergence for sequences is different from the weak convergence) and we prove that if a dual Banach space Z is a kappa-space with the approximation property, then the uniform algebra A(B) on the unit ball of Z generated by the weak-star continuous polynomials is not tight., DGICYT, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we show that the Aron-Berner type extension of polynomials preserves the P-continuity property. To this end we introduce a new version of Goldstine's Theorem for locally complemented subspaces., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

In order to obtain global inversion theorems for mappings between length metric spaces, we investigate sufficient conditions for a local homeomorphism to be a covering map in this context. We also provide an estimate of the domain of invertibility of a local homeomorphism around a point, in terms of a kind of lower scalar derivative. As a consequence, we obtain an invertibility result using an analog of the Hadamard integral condition in the frame of length spaces. Some applications are given to the case of local diffeomorphisms between Banach-Finsler manifolds. Finally, we derive a global inversion theorem for mappings between stratified groups., D.G.E.S. (Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

It is shown that, in contrast to the complex case, every paracompact finite-dimensional real-analytic manifold can be embedded as a bounded and closed real-analytic submanifold of every infinite-dimensional real Banach space., CAICYT, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

For an infinite-dimensional Riemannian manifold M we denote by C1b(M) the space of all real bounded functions of class C(1) on M with bounded derivative. In this paper we shall see how the natural structure of normed algebra on C1b(M) characterizes the Riemannian structure of M, for the special case of the so-called uniformly bumpable manifolds. For that we need, among other things, to extend the classical Myers-Steenrod theorem on the equivalence between metric and Riemannian isometries, to the setting of infinite-dimensional Riemannian manifolds., D.G.I. (Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

Several forms of the Dunford-Pettis property are studied, each related to a different mode of sequential convergence, and a different class of weakly compact functions. The relationship between these Dunford-Pettis properties is investigated, and the appearance of previously studied Dunford-Pettis properties is pointed out, giving a unifying approach to the subject., DGICYT (Ministerio de Educación y Cultura de España), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We relate the moduli of asymptotic uniform smoothness and convexity of a Banach space with the existence of upper and lower l(p)-estimates of sequences in the space. To this end, we introduce two properties which are related to the (m(p))-property defined by Kalton and Werner. In this way we obtain a connection between the moduli of asymptotic uniform smoothness and convexity, and compactness or weak-sequential continuity of multilinear mappings. Finally, we give some applications to the existence of analytic and asymptotically flat norms on a Banach space., D.G.I. (Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

The concept of subdifferentiability is studied in the context of C-1 Finsler manifolds (modeled on a Banach space with a Lipschitz C-1 bump function). A class of Hamilton-Jacobi equations defined on C-1 Finsler manifolds is studied and several results related to the existence and uniqueness of viscosity solutions are obtained., DGES (Spain), MEC, CONICYT-Chile through FONDECYT, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

In order to find metric spaces X for which the algebra Lip*(X) of bounded Lipschitz functions on X determines the Lipschitz structure of X, we introduce the class of small-determined spaces. We show that this class includes precompact and quasi-convex metric spaces. We obtain several metric characterizations of this property, as well as some other characterizations given in terms of the uniform approximation and the extension of uniformly continuous functions. In particular we show that X is small-determined if and only if every uniformly continuous real function on X can be uniformly approximated by Lipschitz functions., MTM2006-03531 ; MTM2004-07665-C02-02 (Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We study the extension of bilinear forms from a given subspace of an L1 -space to the whole space. Precisely, an isomorphic embedding j: E → X is said to be (linearly) 2-exact if bilinear forms on E can be (linear and continuously) extended to X through j . We present some necesary and some sufficient conditions for an embedding j: E → X to be 2-exact when X is an L1 -space., DGICYT, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

For a metric space X, we study the space D(infinity)(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D(infinity)(X) is compared with the space LIP(infinity)(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D(infinity)(X) with the Newtonian-Sobolev space N(1,infinity)(X). In particular, if X Supports a doubling measure and satisfies a local Poincare inequality, we obtain that D(infinity)(X) = N(1,infinity)(X)., DGES (Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

DGICYT, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We first study the reflexivity of the space P(E-m,F) of continuous m-homogeneous polynomials between Banach spaces E and F. Then, in a more general way, we obtain conditions under which the spaces P(E-m,F)(n) and P(E-m",F") are canonically isomorphic., DGICYT (Spain), CNPq grant (Brasil), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We introduce, study and characterize orthogonally additive holomorphic functions f:U -> a", where U is an open subset of C(K). We are led to consider orthogonal additivity at different points of U., Ministerio de Ciencia e Innovación, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

Proceedings of the Seventh Italian-Spanish Conference of General Topology and its Applications, Badajoz (Spain), September 7-10, 2010., Our aim in this note is to give an extension of the classical Myers-Nakai theorem in the context of Finsler manifolds. To achieve this, we provide a general result in this line for subalgebras of bounded Lipschitz functions on length metric spaces. We also establish some connection with the uniform approximation of bounded Lipschitz functions by functions in the subalgebra, keeping control on the Lipschitz constants, D.G.I., UCLA (Venezuela), Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

We study the existence of everywhere differentiable functions which are almost everywhere solutions of quite general Hamilton-Jacobi equations on open subsets of R(d) or on d-dimensional manifolds whenever d >= 2. In particular, when M is a Riemannian manifold, we prove the existence of a differentiable function a on M which satisfies the Eikonal equation parallel to del u(x)parallel to(x) = 1 almost everywhere on M., Ministerio de Ciencia e Innovación (MICINN), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

On every real Banach space X we introduce a locally convex topology tau(p), canonically associated to the weak-polynomial topology w(P). It is proved that tau(p) is the finest locally convex topology on X which is coarser than w(P). Furthermore, the convergence of sequences is considered, and sufficient conditions on X are obtained under which the convergent sequences for w(P) and for tau(P) either coincide with the weakly convergent sequences (when X has the Dunford-Pettis property) or coincide with the norm-convergent sequences (when X has nontrivial type)., DGES (Spain), Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

In this note we prove that the uniformity of a complete metric space X is characterized by the vector lattice structure of the set U(X) of all uniformly continuous real functions on X., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Let f be a split submersion between paracompact Banach manifolds. We obtain here various conditions for f to be a fiber bundle. First, we give general conditions in terms of path-liftings. As a consequence, we deduce several criteria: For example, f is a fiber bundle provided it satisfies either some topological requirements ( such as being a proper or a closed map) or, in the case of Finsler manifolds, some metric requirements (such as Hadamard integral condition)., CONACyT (México), DGES (Spain, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We investigate infinite dimensional Banach spaces equipped with the initial topology with respect to the continuous polynomials. We show nonlinear properties for this topology in both the real and the complex case. A new property for Banach spaces, polynomial Dunford-Pettis property, is introduced. For spaces with this property the compact sets in the topology induced by the polynomials are shown to be invariant under the summation map. For most real Banach spaces we characterize the polynomially compact sets as the bounded sets that are separated from zero by the positive polynomials., DGICYT (Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

In this note we prove that if a differentiable function oscillates between y« and « on the boundary of the unit ball then there exists a point in the interior of the ball in which the differential of the function has norm equal or less than« . This kind of approximate Rolle’s theorem is interesting because an exact Rolle’s theorem does not hold in many infinite dimensional Banach spaces. A characterization of those spaces in which Rolle’s theorem does not hold is given within a large class of Banach spaces. This question is closely related to the existence of C1 diffeomorphisms between a Banach space X and X _ _04 which are the identity out of a ball, and we prove that such diffeomorphisms exist for every C1 smooth Banach space which can be linearly injected into a Banach space whose dual norm is locally uniformly rotund (LUR)., Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

In this paper we study real lattice homomorphisms on a unital vector lattice L subset of C(X), where X is a completely regular space. We stress on topological properties of its structure spaces and on its representation as point evaluations. These results are applied to the lattice L = Lip(X) of real Lipschitz functions on a metric space. Using the automatic continuity of lattice homomorphisms with respect to the Lipschitz norm, we are able to derive a Banach-Stone theorem in this context. Namely, it is proved that the unital vector lattice structure of Lip (X) characterizes the Lipschitz structure of the complete metric space X. In the case L = Lip (X) of bounded Lipschitz functions, an analogous result is obtained in the class of complete quasiconvex metric spaces., DGES, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We study the extension of bilinear and multilinear forms from a given subspace of a Banach space to the whole space. Precisely, an isomorphic embedding j : E --> X is said to be (linearly) N-exact if N-linear forms on E can be (linear and continuously) extended to X through j. We present some necessary and sufficient conditions for j to be 2-exact, as well as several examples of 2-exact embeddings. We answer a problem of Zalduendo: in a cotype 2 space bilinear extendable and integral forms coincide., DGICYT, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We present an example showing that a classical result due to Glaeser about the closedness of composition subalgebras of infinitely differeuniable functions cannot be extended to the case of weakly uniformly differentiable functions on Banach spaces, CAICYT, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We study the global invertibility of non-smooth, locally Lipschitz maps between infinite-dimensional Banach spaces, using a kind of Palais-Smale condition. To this end, we consider the Chang version of the weighted Palais-Smale condition for locally Lipschitz functionals in terms of the Clarke subdifferential, as well as the notion of pseudo-Jacobians in the infinite-dimensional setting, which are the analog of the pseudo-Jacobian matrices defined by Jeyakumar and Luc. Using these notions, we derive our results about existence and uniqueness of solution for nonlinear equations. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context., MICINN, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), FALSE, pub

For a large class of metric spaces with nice local structure, which includes Banach-Finsler manifolds and geodesic spaces of curvature bounded above, we give sufficient conditions for a local homeomorphism to be a covering projection. We first obtain a general condition in terms of a path continuation property. As a consequence, we deduce several conditions in terms of path- liftings involving a generalized derivative, and in particular we obtain an extension of Hadamard global inversion theorem in this context. Next we prove that, in the case of quasi-isometric mappings, some of these sufficient conditions are also necessary. Finally, we give an application to the existence of global implicit functions., Promep (México), D.G.E.S. (Spain), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We are concerned here with Sobolev-type spaces of vector-valued functions. For an open subset Ω ⊂ R N and a Banach space V , we compare the classical Sobolev space W1,p(Ω, V ) with the so-called Sobolev-Reshetnyak space R1,p(Ω, V ). We see that, in general, W1,p(Ω, V ) is a closed subspace of R1,p(Ω, V ). As a main result, we obtain that W1,p(Ω, V ) = R1,p(Ω, V ) if, and only if, the Banach space V has the Radon-Nikod´ym property., Ministerio de Ciencia, Innovación y Universidades (España), Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, FALSE, pub

For each quasi-metric space X we consider the convex lattice SLip(1)(X) of all semi-Lipschitz functions on X with semi-Lipschitz constant not greater than 1. If X and Y are two complete quasi-metric spaces, we prove that every convex lattice isomorphism T from SLip(1)(Y) onto SLip(1)(X) can be written in the form Tf = c . (f o tau) + phi, where tau is an isometry, c > 0 and phi is an element of SLip(1)(X). As a consequence, we obtain that two complete quasi-metric spaces are almost isometric if, and only if, there exists an almost-unital convex lattice isomorphism between SLip(1)(X) and SLip(1) (Y)., Ministerio de Economía y Competitividad (MINECO), Junta de Extremadura, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

Given a surjective mapping f : E -> F between Banach spaces, we investigate the existence of a subspace G of E, with the same density character as F, such that the restriction of f to G remains surjective. We obtain a positive answer whenever f is continuous and uniformly open. In the smooth case, we deduce a positive answer when f is a C-1-smooth surjection whose set of critical values is countable. Finally we show that, when f takes values in the Euclidean space R-n, in order to obtain this result it is not sufficient to assume that the set of critical values of f has zero-measure., Ministerio de Economía y Competitividad (MINECO), Academy of Finland, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We prove that a locally complete metric space endowed with a doubling measure satisfies an infinity-Poincare inequality if and only if given a null set, every two points can be joined by a quasiconvex curve which "almost avoids" that set. As an application, we characterize doubling measures on R satisfying an infinity-Poincare inequality. For Ahlfors Q-regular spaces, we obtain a characterization of p-Poincare inequality for p > Q in terms of the p-modulus of quasiconvex curves connecting pairs of points in the space. A related characterization is given for the case Q - 1 < p <= Q.

We study the global inversion of a continuous nonsmooth mapping f : R-n -> R-n, which may be non-locally Lipschitz. To this end, we use the notion of pseudo-Jacobian map associated to f, introduced by Jeyakumar and Luc, and we consider a related index of regularity for f. We obtain a characterization of global inversion in terms of its index of regularity. Furthermore, we prove that the Hadamard integral condition has a natural counterpart in this setting, providing a sufficient condition for global invertibility., MINECO Project (Spain), CONICYT-Chile through FONDECYT project, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete C-k Finsler manifold M is determined by the normed algebra C-b(k)(M) of all real-valued, bounded and C-k smooth functions with bounded derivative defined on M. As a consequence, we obtain: (i) the Finsler structure of a finite-dimensional and complete C-k Finsler manifold M is determined by the algebra C-b(k)(M); (ii) the weak Finsler structure of a separable and complete C-k Finsler manifold M modeled on a Banach space with a Lipschitz and C-k smooth bump function is determined by the algebra C-b(k)(M); (iii) the weak Finsler structure of a C-1 uniformly bumpable and complete C-1 Finsler manifold M modeled on a Weakly Compactly Generated (WCG) Banach space is determined by the algebra C-b(1)(M); and (iv) the isometric structure of a WCG Banach space X with a C-1 smooth bump function is determined by the algebra C-b(1)(X)., DGES (Spain) Project, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

Let f : E -> F be a surjective mapping between two real or complex Banach spaces, with f having some strong differentiability properties. We investigate when there is a smaller Banach space G subset of E such that the restriction of f to G remains surjective., MICINN, NSERC, Canada Research Chairs program, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), FALSE, pub

We study the smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. We prove that, given a Lipschitz function f : M -> R defined on a connected, second countable Finsler manifold M, for each positive continuous function epsilon : M -> (0, infinity) and each r > 0, there exists a C-1-smooth Lipschitz function g : M -> R such that vertical bar f(x) - g(x)vertical bar <= epsilon(x), for every x is an element of M, and Lip(g) <= Lip(f) + r. As a consequence, we derive a completeness criterium in the class of what we call quasi-reversible Finsler manifolds. Finally, considering the normed algebra C-b(1)(M) of all C-1 functions with bounded derivative on a complete quasi-reversible Finsler manifold M, we obtain a characterization of algebra isomorphisms T : C-b(1)(N) -> C-b(1)(M) as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds., D.G.I. (Spain) Grant, (CDCHT-UCLA) (Venezuela), Depto. de Álgebra, Geometría y Topología, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

We consider the problem of finding sufficient conditions for a locally Lipschitz mapping between Finsler manifolds to be a global homeomorphism. For this purpose, we develop the notion of Clarke generalized differential in this context and, using this, we obtain a version of the Hadamard integral condition for invertibility. As consequences, we deduce some global inversion and global injectivity results for Lipschitz mappings on R-n in terms of spectral conditions of its Clarke generalized differential., MICINN (Spain), Fundacion Cajamadrid, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, Instituto de Matemática Interdisciplinar (IMI), TRUE, pub

DGES (Spain); Taft Foundation of the University of Cincinnati, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub