Cones with bounded and unbounded bases and reflexivity

Enrico Miglierina, E. Casini, E. Miglierina

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)

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 ℓ^1 of ℓ^1 is a mixed based cone and that every mixed based cone contains a conically isomorphic copy of ℓ^1_+. 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 ℓ^1 and c_0 in a Banach space.
Original languageEnglish
Pages (from-to)2356-2366
Number of pages11
JournalNONLINEAR ANALYSIS
Volume72
DOIs
Publication statusPublished - 2010

Keywords

  • Based cone
  • Cone conically isomorphic to l^1_+
  • Reflexive space
  • Strongly summing sequence

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