LATTICE OPERATIONS OF POSITIVE BILINEAR MAPPINGS

In this paper we establish extension theorems for additive mappings ϕ : A+ × B+ → C+, where A, B are Riesz spaces (lattice ordered spaces or vector lattices) and C is an order complete Riesz space, to the whole of A × B, thereby extending well-known results for additive mappings between Riesz spaces. We prove, in particular, that when A, B and C are order complete Riesz spaces, the ordered vector space Bb(A × B,C) of all order bounded bilinear mappings has the structure of a lattice space.