Banach-Kantorovich spaces
Let E be an ordered complete vector lattice and X be an arbitrary vector space. A vector-valued norm \(p: X\to E\) is said to be decomposable if for any \(x\in X\) the representation \(p(x)=e_ 1+e_ 2\) with disjoint \(e_ 1,e_ 2\in E\) implies that there exist \(x_ 1,x_ 2\in X\) such that \(x=x_ 1+x_ 2\) and \(p(x_ i)=e_ i\), \(i=1,2\). A Banach-Kantorovich space (or briefly BK-space) is a pair (X,p) where p is a decomposable E- valued norm on X and X is complete with respect to order convergence in E. The main result of the paper states that there is a natural correspondence between BK-spaces and Banach spaces in the suitable Boolean-valued models of set theory. This fact explains the parallelism of the theories of BK-spaces and Banach spaces. On the other hand, it gives a transfer principle in BK-spaces theory.
complete with respect to order convergence
scientific article; zbMATH DE number 3921250
scientific article