In this paper we prove two characterizations of reflexivity for a Banach space X. The first one is based on the existence in X of a closed convex cone with nonempty interior such that all the bases generated by a strictly positive functional are bounded, while the second one is stated in terms of non existence of a cone such that has bounded and unbounded bases (both generated by strictly positive functionals) simultaneously. We call such a cone mixed based cone. We study the features of this class of cones. In particular, we show that every cone conically isomorphic to the nonnegative orthant ℓ1+ of ℓ1 is a mixed based cone and that every mixed based cone contains a conically isomorphic copy of ℓ1+. Moreover we give a detailed description of the structure of a mixed based cone. This approach allows us to prove some results concerning the embeddings of ℓ1 and c0 in a Banach space.

Cones with bounded and unbounded bases and reflexivity

Casini, E.;Miglierina, E.
2010

Abstract

In this paper we prove two characterizations of reflexivity for a Banach space X. The first one is based on the existence in X of a closed convex cone with nonempty interior such that all the bases generated by a strictly positive functional are bounded, while the second one is stated in terms of non existence of a cone such that has bounded and unbounded bases (both generated by strictly positive functionals) simultaneously. We call such a cone mixed based cone. We study the features of this class of cones. In particular, we show that every cone conically isomorphic to the nonnegative orthant ℓ1+ of ℓ1 is a mixed based cone and that every mixed based cone contains a conically isomorphic copy of ℓ1+. Moreover we give a detailed description of the structure of a mixed based cone. This approach allows us to prove some results concerning the embeddings of ℓ1 and c0 in a Banach space.
Reflexive space; based cone; cone conically isomorphic to ℓ1+; strongly summing sequence
Casini, E.; Miglierina, E.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11383/1712109
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