Necessary and Sufficient Conditions for the Order‐ Completeness of Partially Ordered Vector Spaces

Abstract

The HAHN-BANACH-theorem is known to have fundamental importance for several fields of mathematics. This theorem is not used concerning functionals, but operators which map into a real partially ordered vector space.In this paper is shown that the validity of this theorem is equivalent t o the validity of various well known propositions of mathematical programming and of partially ordered vector spaces and of operator-inequalities. Among these propositions, for instance, we find the F~~~a s -M~~~o w s~~-t h e o r e m , researched in such equivalences for functionals.If we proved these equivalences, we could give sufficient conditions for the validity of theorems of mathematical programming. On the other hand, by including To's result [S], we reach statements which characterize the order completeness of partially ordered vector spaces.With this paper, we have continued and completed our paper [3] using the same notations.

Hahn-Banach and Sandwich Theorems for Equivariant Vector Lattice-Valued Operators and Applications

“…In this paper, we extend to linear and equivariant vector lattice-valued operators the earlier results proved in [23,[47][48][49] in the linear case and in [6] in the invariant setting on Hahn-Banach, sandwich and extension theorems, Fenchel-type duality theorems, Moreau-Rockafellar formula, subdifferential calculus, Farkas and Kuhn-Tucker theorems on convex optimization under constraints. Furthermore, we prove the equivalence of given theorems with amenability.…”

Strong Vector Minimization and Duality

“…Moreover, the KUHN-TUCKER conditions may hold without ordcr-completeness of F, if other parts of the problem are suitably restricted. Also duality and related tlicorcnis iiiay hold, as shown below, when (unlike [$I, [9]) different ordering cones are used for the given (primal) pi*obIcni and its vector dual.…”

Separation of Two Sets in a Product Space

“…It is known that the HAHN-BAxACH-Theorem (without the assumption of the (lubp) of F) follows from each of Theorem 1, Theorem 3, Theorem 4 andTheorem 5 where for such conclusions the (lubp) of F is also not used (see [5], [6], [S]) Therefore, the least upper bound property of F is necessary for each of these statements.…”