Spaces with Dual Order-Isomorphic to ℓ1

We establish that an ordered Banach space is order-isomorphic to c₀ if and only if it is a -Dedekind complete vector lattice and its norm dual is order-isomorphic to ₁.     

Finite-dimensional lattice-subspaces of

Let be linearly independent positive functions in , let be the vector subspace generated by the and let denote the curve of determined by the function , where . We establish that is a vector lattice under the induced ordering from if and only if there exists a convex polygon of with vertices containing the curve and having its vertices in the closure of the range of . We also present an algorithm which determines whether or not is a vector lattice and in case is a vector lattice it constructs a positive basis of . The results are also shown to be valid for general normed vector lattices.

     

Cone characterization of Grothendieck spaces and Banach spaces containing <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>

In this article we study the embeddability of cones in a Banach space X. First we prove that c ₀ is embeddable in X if and only if its positive cone c₀⁺{c_0^+} is embeddable in X and we study some properties of Banach spaces containing c ₀ in the light of this result. So, unlike with the positive cone of ℓ ₁ which is embeddable in any non-reflexive space, c₀⁺{c_0^+} has the same behavior as the whole space c ₀. In the second part of this article we give a characterization of Grothendieck spaces X according to the geometry of cones of X*. By these results we give a partial positive answer to a problem of J.H. Qiu concerning the geometry of cones.     

Demand functions and reflexivity

In the theory of ordered spaces and in microeconomic theory two important notions, the notion of the base for a cone which is defined by a continuous linear functional and the notion of the budget set are equivalent. In economic theory the maximization of the preference relation of a consumer on any budget set defines the demand correspondence which at any price vector indicates the preferred vectors of goods and this is one of the fundamental notions of this theory. Contrary to the finite-dimensional economies, in the infinite-dimensional ones, the existence of the demand correspondence is not ensured. In this article we show that in reflexive spaces (and in some other classes of Banach spaces), there are only two classes of closed cones, i.e. cones whose any budget set is bounded and cones whose any budget set is unbounded. Based on this dichotomy result, we prove that in the first category of these cones the demand correspondence exists and that it is upper hemicontinuous. We prove also a characterization of reflexive spaces based on the existence of the demand correspondences.     

Weakly compact subsets ofL 1(μ,X) andbvca(Σ,X)

In this paper we study the weakly compact subset ofL ₁(μ,X) and bvca(Σ,X), supposing that proper subsets ofX andX ^* have the Radom-Nikodym property.     

Lattice-Subspaces and Positive Bases in Function Spaces

Positivity - Let x 1,...,xn be linearly independent, positive elements of the space RΩ of the real valued functions defined on a set Ω and let X be the vector subspace of RΩ...     

Reflexive cones

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.     

Atomic sublattices and basic derivatives in finance

Positivity - Suppose that E is a vector lattice where the ordering and the lattice operations in E are defined pointwise by a countable family $${\mathcal {F}}=\{f_i|i\in {{\mathbf {N}}}\}$$ of...     

Embeddability of L1(μ) in dual spaces, geometry of cones and a characterization of c0

In this article we suppose that (Ω,Σ,μ) is a measure space and T an one-to-one, linear, continuous operator of L₁(μ) into the dual E′ of a Banach space E. For any measurable set A consider the image T(L⁺₁(μ_A)) of the positive cone of the space L₁(μ_A) in E′, where μ_A is the restriction of the measure μ on A. We provide geometrical conditions on the cones T(L⁺₁(μ_A)) which yield that the measure μ is atomic, i.e., that L₁(μ) is lattice isometric to , where denotes the set of atoms of μ. This result yields also a new characterization of c₀(Γ).     

The Structure of Lattice-Subspaces

In Polyrakis (1983; Math. Proc. Cambridge Phil. Soc. 94, 519) it is proved that each infinite-dimensional, closed lattice-subspace of . . .₁ is order-isomorphic to . . .₁ and in Polyrakis (1987; Math. Anal. Appl. 184, 1) that each separable Banach lattice is order isomorphic to a closed lattice-subspace of C[0,1]. Therefore . . .₁ contains only one lattice-subspace but C[0,1] contains all the separable Banach lattices. In the first section of this article we study the kind of the order embeddability of a separable Banach lattice in C[0,1]. We show that the AM spaces have the ``best' behavior and the AL-spaces the ``worst'. In the second section we prove that the closure of a lattice-subspace is not necessarily a lattice-subspace and in the least one we study lattice-subspaces with positive bases.