Francisco Javier García-Pacheco

Some questions about rotundity and renormings in Banach spaces

In this paper, we show some results involving classical geometric concepts. For example, we characterize rotundity and Efimov-Stechkin property by mean of faces of the unit ball. Also, we prove that every almost locally uniformly rotund Banach space is locally uniformly rotund if its norm is Frechet differentiable. Finally, we also provide some theorems in which we characterize the (strongly) exposed points of the unit ball using renormings. 2000 Mathematics subject classification: primary 46B20.

A short note about exposed points in real banach spaces

Abstract We express the set of exposed points in terms of rotund points and non-smooth points. As long as we have Banach spaces each time “bigger”, we consider sets of non-smooth points each time “smaller”.

VECTOR SUBSPACES OF THE SET OF NON-NORM-ATTAINING FUNCTIONALS

An example is found of a nonreflexive Banach space X such that the union of {0} and the set of non-norm-attaining functionals on X contains no two-dimensional subspace.

Matrix summability and uniform convergence of series

. Some classical results about uniform convergence of unconditionally convergent series are generalized to weakly unconditionally Cauchy series by means of the matrix summability method for regular matrices.

Geometric properties on non-complete spaces

Abstract The purpose of this paper is to study certain geometrical properties for non-complete normed spaces. We show the existence of a non-rotund Banach space with a rotund dense maximal subspace. As a consequence, we prove that every separable Banach space can be renormed to be non-rotund and to contain a dense maximal rotund subspace. We then construct a non-smooth Banach space with a dense maximal smooth subspace. We also study the Krein-Milman property on non-complete normed spaces and provide a sufficient condition for an infinite dimensional Banach space to have an infinite dimensional, separable quotient.

Geometry of isometric reflection vectors

In this paper we study the geometry of isometric reflection vectors. In particular, we generalize known results by proving that the minimal face that contains an isometric reflection vector must be an exposed face. We also solve an open question by showing that there are isometric reflection vectors in any two dimensional subspace that are not isometric reflection vectors in the whole space. Finally, we prove that the previous situation does not hold in smooth spaces, and study the orthogonality properties of isometric reflection vectors in those spaces.

A NOTE ON L 2 -SUMMAND VECTORS IN DUAL SPACES

It is shown that every L 2 -summand vector of a dual real Banach space is a norm-attaining functional. As consequences, the L 2 -summand vectors of a dual real Banach space can be determined by the L 2 -summand vectors of its predual; for every n ∈ , every real Banach space can be equivalently renormed so that the set of norm-attaining functionals is n-lineable; and it is easy to find equivalent norms on non-reflexive dual real Banach spaces that are not dual norms.

Rotundity in transitive and separable Banach spaces

In every separable Banach space the set of smooth points of the unit ball is a Gδ dense subset of the unit sphere. In this paper, we find some conditions in order to obtain a similar result for rotund points. For instance, we prove that if the unit ball of a smooth and separable Banach space is free of rotund points then the set of non-norm-attaining functionals of norm 1 is residual in the unit sphere of the dual. Furthermore, by taking profit of these techniques we provide positive approaches to the Banach-Mazur conjecture for rotations.

Linear topologies and sequential compactness in topological modules

Abstract We prove that an absolute semi-valued ring is first-countable if the set of invertibles is separable and its closure contains 0. We also show that every linearly topologized topological module over an absolute semi-valued ring whose invertibles approach 0 has the trivial topology. We also show that every sequentially compact set in a topological module is bounded if the module is over an absolute semi-valued ring whose set of invertibles is separable and its closure contains 0. Finally, we find sufficient conditions for a sequentially compact neighborhood of 0 to force the corresponding module to be finitely generated.

Balanced and absorbing subsets with empty interior

Abstract Our first result says that every real or complex infinite-dimensional normed space has an unbounded absolutely convex and absorbing subset with empty interior. As a consequence, a real normed space is finite-dimensional if and only if every convex subset containing 0 whose linear span is the whole space has non-empty interior. In our second result we prove that every real or complex separable normed space with dimension greater than 1 contains a balanced and absorbing subset with empty interior which is dense in the unit ball. Explicit constructions of these subsets are given.