Smoothness in weakly compactly generated Banach spaces

Approximation by functions with bounded derivative on Banach spaces

“…A version of this theorem was later shown to hold for general reflexive spaces in the key paper [9] by Torunczyk, and both these results were generalised to weakly compactly generated spaces in [5].…”

Smooth Approximations in Banach Spaces

“…This proves (b). The more general statement (a) follows because the above renorming of C(K) and transfer techniques as used in [3,8,9] show that X has an LUC norm whose dual is also LUC. One obtains (c) from the recent papers [6,7], which show that X has an LUC norm whose dual is LUC whenever X* is WCD.…”

“…One obtains (c) from the recent papers [6,7], which show that X has an LUC norm whose dual is LUC whenever X* is WCD. We remark that for X* a subspace of a WCG space the full generality of [6,7] is not needed the results of [8,9] suffice. Also, when X* is WCG, (c) is a special case of (a) with KM = 0.…”

“…Usually such embeddings of X into co(r) are constructed with the help of a linear injection of X into Co(Y). In this direction, the combined efforts of [8] and [13] show that X admits Ck-smooth partitions of unity whenever X has a Ck-smooth bump function and X or X* is WCG. However, C1-smooth partitions of unity on X are obtained here by the purely geometric condition that X admits an equivalent LUC norm whose dual is also LUC.…”

The Mazur property for compact sets

“…In this section we will prove a "transfer theorem" which is analogous to the corresponding one for property (/) [2]. For other "transfer theorems" see [4], [5].…”