The Hahn-Banach theorem for finite dimensional spaces

Abstract

1. Introduction. An immediate generalization of the Hahn-Banach Theorem on the dominated extension of linear functionals is obtained by replacing the real number field by a boundedly complete vector lattice as the range space of the function. The question arises: Do boundedly complete vector lattices characterize those range spaces which permit dominated extensions? It is proved in [1], [4] that an ordered linear space is a boundedly complete vector lattice if and only if both the positive wedge is lineally closed and the Hahn-Banach Theorem holds with the space taken as the range of the extensions. Thus, the above question has an affirmative answer if the positive wedge must be closed in a space for which the Hahn-Banach Theorem is valid. This paper shows that at least for finite dimensional ordered linear spaces this is indeed the case. An example is presented in [4] of a two dimensional ordered linear space whose positive wedge is not lineally closed and it is erroneously asserted that this space permits Hahn-Banach type extensions. The error in the argument is that the extensions described may not be well defined. ?2 introduces preliminary definitions and theorems. In ?3, finite dimensional boundedly complete vector lattices are characterized by the existence of a partly positive basis for the positive wedge. This characterization is a slight generalization of a theorem of Yudin [5] or Nagy [3]. In ?4 it is proved that if a finite dimensional ordered linear space permits Hahn-Banach type extensions, then the closure of the positive wedge determines an ordered linear space which is a boundedly complete vector lattice. This restricts the class of positive wedges which must be considered in ascertaining the class which admits dominated extensions. Examples are given in ?5 which are used in ?6 to complete the proof that a finite dimensional ordered linear space admits dominated extensions if and only if the space is a boundedly complete vector lattice.