A representation of sup-completion

Abstract

It was showed by Donner in [Extension of positive operators and Korovkin theorems, Lecture Notes in Mathematics, vol. 904, Springer-Verlag, Berlin-New York, 1982] that every order complete vector lattice X X may be embedded into a cone X s X^s , called the sup-completion of X X . We show that if one represents the universal completion of X X as C ∞ ( K ) C^\infty (K) , then X s X^s is the set of all continuous functions from K K to [ − ∞ , ∞ ] [-\infty ,\infty ] that dominate some element of X X . This provides a functional representation of X s X^s , as well as an easy alternative proof of its existence.