An extension of a structure theorem of Bourgain

AbstractBourgain has shown that if X is a Banach space which does not have the Radon-Nikodym Property (respectively, the Point-of-Continuity Property), then X has a subspace Y which also fails to have the respective property, but which has a finite dimensional Schauder decomposition. We extend this result to spaces X failing the Convex Point-of-Continuity Property

On the Lambert W Function

The Lambert W function is defined to be the multivalued inverse of the function w 7! we w . It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches of W , an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing W . On the Lambert W function 2 1. Introduction In 1758, Lambert [47] solved the trinomial equation x = q + x m by giving a series development for x in powers of q. Later [48], he extended the series to give powers of x as well. Euler [28] transformed Lambert's equation into a more symmetric form by substituting x \Gammafi for x and setting m = fffi and q = (ff \Gamma fi)v. His version of the equation was x ff \Gamma x fi = (ff \Gamma fi)vx ff+fi ; (1:1) and his version of Lambert's series solution was x n = 1 + nv + 1 2 ..