, Volume 17, Issue 3, pp 911-933

Reflexive cones

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Abstract

Reflexive cones in Banach spaces are cones with weakly compact intersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of \(\ell _{1}\) . Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those generated by a Schauder basis and the reflexive cones regarded as ordering cones in Banach spaces. Finally, it is worth to point out that a characterization of reflexive spaces and also of the spaces with the Schur property by the properties of reflexive cones is given.

I.A. Polyrakis and F. Xanthos have been co-financed by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: Heracleitus II. Investing in Knowledge society through the European Social Fund.