On the duality mapping sets in abstract \(M\)-spaces. (Q1866451)

The authors characterize various properties associated with support functionals in abstract \(M\) spaces. For a normed space \(X\), the duality mapping set \(J(x)\) associated with an element \(x\in X\) is defined as the set of functionals \(f^*\) in the dual space such that \((f^*(x))=\| x\| ^2=\| f^*\| ^2\) . For abstract \(M\) spaces, characterizations of properties such as compactness of the duality mapping set \(J(x)\) in terms of decompositions of the element \(x\) are provided. For example, if an abstract \(M\) space is Dedekind complete and order semi-continuous then \(J(x)\) is compact if and only if \(x = y + z\) where \(y\) and \(z\) are disjoint, \(\| z\| <\| x\| \) and \(y\) can be appropriately decomposed into atoms.