Insertion theorems for maps to Banach lattices
Semicontinuous maps from topological spaces into vector lattices are investigated. For a topological space \(X\) and an ordered topological vector space \(Y\) we say that \((X,Y)\) has the insertion property if for every upper semicontinuous map \(g: X\to Y\) and every lower semicontinuous map \(h: X\to Y\) with \(g\leq h\), there exists a continuous map \(f: X\to Y\) such that \(g\leq f\leq h\). In the paper there are several insertion theorems for maps from a topological space to Banach lattices. Similarly, for a topological space \(X\) and an ordered topological vector space \(Y\), we say that \((X,Y)\) has the one-side insertion property if for every upper semicontinuous map \(g: X\to Y\) there exists a continuous map \(f: X\to Y\) such that \(g\leq f\) or for every lower-continuous map \(h: X\to Y\) there exists a continuous map \(f: X\to Y\) such that \(f\leq h\). In the paper there are also results on one-side insertion property for maps from topological space to Banach lattices.
scientific article; zbMATH DE number 5724590
scientific article