Sublinear Functionals and Ideals in Partially Ordered Vector Spaces

Abstract

1. LET F be a vector space over the real field R. By a sublinear functional^ on V we mean a real-valued function p defined on F which is subadditive and positive-homogeneous: i.e. if x, y e F and a ^ 0, then p{x-\\-y) < p{%)-\\-p(y) and p(ax) = <xp(x). The most important property of sublinear functionals was established by S. Banach. T H E O R E M 1 (S. B a n a c h (3), p. 27). Letp be a sublinear functional on a vector space V over R. Let <f>0 be a linear functional defined on a linear sub-space Vo of V with <f>Q{x) < p(x) for all x e Vo. Then there exists a linear functional <f> defined on V with <f>(x) ^ p ( x) for all xeV and <f>(x) = <f>0(x) for all XEVQ. Since we may take the one-dimensional subspace (x0) generated by a point x0 as Vo and <f>0{x0) = p{x0), this theorem shows, in particular, that p is the upper envelope of all linear functionals <f> on V dominated by p, i.e. with <f>(x) < p(x) for all x e V. General sublinear functionals have received little attention, though special types of sublinear functionals (norms and seminorms) have been extensively studied. By a partially ordered vector space we mean a vector space V over R in which a non-empty subset F+, the positive cone, is specified having the following properties: Axiom (i) x, y eV+ and a> 0 imply that x-\\-y e F + and <xx e V+; Axiom (ii) xeV+ and — x e F+ imply that x = 0. As usual, we write x ^ y (or y < x) iix—y e F+, and write x> y (or y < x) if x ^ y and x ^ y. Partially ordered vector spaces in this general sense have also been somewhat neglected, though many mathematicians have studied linear lattices. An ideal J in a partially ordered vector space is a linear subspace / such that x e I whenever there is an element ie I with — i < x < i. In § 2 we