TY - JOUR

T1 - Cones with bounded and unbounded bases and reflexivity

AU - Miglierina, Enrico

AU - Casini, E.

AU - Miglierina, E.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Based cone

KW - Cone conically isomorphic to l^1_+

KW - Reflexive space

KW - Strongly summing sequence

KW - Based cone

KW - Cone conically isomorphic to l^1_+

KW - Reflexive space

KW - Strongly summing sequence

UR - http://hdl.handle.net/10807/1427

UR - http://www.sciencedirect.com/science/article/pii/s0362546x09010979

U2 - 10.1016/j.na.2009.10.036

DO - 10.1016/j.na.2009.10.036

M3 - Article

VL - 72

SP - 2356

EP - 2366

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

ER -