Infimum and domination principles in vector lattices

Peter A. Fowler

Vol. 59 (1975), No. 2, 437–443

DOI: 10.2140/pjm.1975.59.437

Abstract

The main purpose of this paper is to demonstrate (a) that the potential theoretic notions of infimum principle and domination principle are meaningful in a setting of a vector lattice with a monotone map to the dual space, and (b) in this general setting these two principles are equivalent under very weak hypotheses.

THEOREM 1. If T : L → L′ is a strictly monotone map, L a vector lattice, then the infimum principle implies the domination principle.

THEOREM 2. If T : B → B′ is a monotone, coercive hemi-continuous map, B a reflexive Banach space and vector lattice with closed positive cone, then the domination principle implies the infimum principle.

Mathematical Subject Classification 2000

Primary: 46A40

Secondary: 47H99

Milestones

Received: 23 September 1974

Published: 1 August 1975