Antonio Aizpuru

Density by moduli and statistical convergence

AbstractBy 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 modulus function.

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”.

Boolean algebras and uniform convergence of series

Abstract Several classical results on uniform convergence of unconditionally Cauchy series are generalized to weakly unconditionally Cauchy series. This uniform convergence is characterized through subalgebras and subfamilies of P ( N ) . A generalization of the Orlicz–Pettis theorem is also proved by mean of subalgebras of P ( N ) .

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.

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.

ALMOST TRANSITIVITY IN

By means of

\mathcal{C}_0

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SPACES OF VECTOR-VALUED FUNCTIONS

-structure and dimension theory, we generalize some known results and obtain some new ones on almost transitivity in