A Mazur–Orlicz Type Theorem in Interval Analysis and Its Consequences

Abstract

Classical extension theorems for linear functionals or, more generally, for linear operators in the setting of vector spaces are well known. For example, the Hahn–Banach Theorem and the Mazur–Orlicz Theorem extend linear functionals (operators) dominated in a certain sense by sublinear functionals (operators). It is also known that these theorems have many applications. To get more applications we intend to give some versions of these theorems in interval analysis. In the literature of this field, intervals are viewed as an extension of any value that they contain, motivated by the fact that in many practical situations some values are known with interval uncertainty. We will work with intervals in ordered vector spaces. It is known that the set of all closed intervals in such spaces is not a vector space. Indeed, for example, there is no additive inverse element for each closed interval. Therefore certain difficulties arise in the proofs of the extension results.