Minimal lattice-subspaces

In this paper the existence of minimal lattice-subspaces of a vector lattice containing a subset of is studied (a lattice-subspace of is a subspace of which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology on and is -closed (especially if is a Banach lattice with order continuous norm), then minimal lattice-subspaces with -closed positive cone exist (Theorem 2.5).

In the sequel it is supposed that is a finite subset of , where is a compact, Hausdorff topological space, the functions are linearly independent and the existence of finite-dimensional minimal lattice-subspaces is studied. To this end we define the function where . If is the range of and the convex hull of the closure of , it is proved:

(i)

There exists an -dimensional minimal lattice-subspace containing if and only if is a polytope of with vertices (Theorem 3.20).

(ii)

The sublattice generated by is an -dimensional subspace if and only if the set contains exactly points (Theorem 3.7).

This study defines an algorithm which determines whether a finite-dimensional minimal lattice-subspace (sublattice) exists and also determines these subspaces.

     

Hereditary order convexity inL(X,Y)

LetX be a topological vector space,Y an ordered topological vector space andL(X,Y) the space of all linear and continuous mappings fromX intoY. The hereditary order-convex cover [K] ^h of a subsetK ofL(X,Y) is defined by [K] ^h ={A∈L(X,Y):Ax∈[Kx] for allx∈X}, where[Kx] is the order-convex ofKx. In this paper we study the hereditary order-convex cover of a subset ofL(X,Y). We show how this cover can be constructed in specific cases and investigate its structural and topological properties. Our results extend to the spaceL(X,Y) some of the known properties of the convex hull of subsets ofX ^*.