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Lineability of functionals and operators

Francisco Javier García-Pacheco;  Daniele Puglisi — 2010

Studia Mathematica

This article is divided into two parts. The first one is on the linear structure of the set of norm-attaining functionals on a Banach space. We prove that every Banach space that admits an infinite-dimensional separable quotient can be equivalently renormed so that the set of norm-attaining functionals contains an infinite-dimensional vector subspace. This partially solves a question proposed by Aron and Gurariy. The second part is on the linear structure of dominated operators. We show that the...

Lineability and spaceability on vector-measure spaces

Giuseppina Barbieri;  Francisco J. García-Pacheco;  Daniele Puglisi — 2013

Studia Mathematica

It is proved that if X is infinite-dimensional, then there exists an infinite-dimensional space of X-valued measures which have infinite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that c a ( ℬ , λ , X ) ∖ M σ , the measures with non-σ-finite variation, contains a closed subspace. Other considerations concern the space of vector measures whose range is neither closed nor convex. All of those results extend in some sense theorems of Muñoz Fernández et al. [Linear Algebra Appl....

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