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

✍ Scribed by Karl-Heinz Elster; Reinhard Nehse

Publisher
John Wiley and Sons
Year
1978
Tongue
English
Weight
448 KB
Volume
81
Category
Article
ISSN
0025-584X

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✦ Synopsis

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 , a KUHN-TUCKER-theorem and a FENCHEL-theorem of duality. Previously HOANG T u Y [5] 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.

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