Vol. 59, No. 2, 1975

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Infimum and domination principles in vector lattices

Peter A. Fowler

Vol. 59 (1975), No. 2, 437–443

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 Lis a strictly monotone map, L a vector lattice, then the infimum principle implies the domination principle.

THEOREM 2. If T : B Bis 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
Received: 23 September 1974
Published: 1 August 1975
Peter A. Fowler