Asplund functions and projectional resolutions of the identity (Q2752993)

Asplund spaces are well-known to have many equivalent characterizations. Recently W. K. Tang succeeded in finding a list of equivalent properties for some continuous convex functions on a Banach space, that are similar to those shared by all continuous convex functions on an Asplund space. A function with these properties was called an Asplund function. More specifically, if \(X\) is an arbitrary Banach space, the continuous convex function \(f\) on \(X\) is an Asplund function if every continuous convex function, majorized by \(f\), is Fréchet differentiable at the points of a dense \(G_\delta\) set. A Banach space is Asplund if and only if its norm is an Asplund function. NEWLINENEWLINENEWLINEThere are analogues to several of the equivalent conditions for Asplund space in terms of Asplund functions. The author develops this theory by finding the analogues of assertions relating Asplund spaces to the projectional resolutions of the identity (PRI). He proves that if \(f\) is an Asplund function on a Banach space \(X\), then \(Y=\overline{sp} \partial f(X)\) has a PRI. Furthermore, if \(X\) is weakly Lindelöf determined (WLD), then \(X\) admits a PRI such that the adjoint projections restricted to \(Y\) form a PRI on \(Y\); also \(X^*\) admits an equivalent dual norm whose restriction to \(Y\) is locally uniformly rotund. Approximation of some Asplund functions on a WLD space is discussed. A list of open problems and a characteristic example is added.