Properties of the Holmes space
Let \((X, d)\) be a metric space and \(m_0 \in X\). Let \(\text{Lip}_0(X)\) denote the space of all real valued Lipschitz functions which vanish at \(m_0\). The Lipschitz free space over \(X\), denoted by \({\mathcal F}(X)\), is the canonical predual of \(\text{Lip}_0(X)\). Let \(X = U\), the Urysohn metric space and fix a point \(m_0 \in U\). Then \({\mathcal H}= {\mathcal F}(U)\) is called the Holmes space and \(\mathcal H\) is universal in the class of all separable Banach spaces. It is not known whether \(\mathcal H\) has a basis. The authors in this paper prove the following properties of \(\mathcal H\): {\parindent=7mm \begin{itemize}\item[(i)] \(\mathcal H\) has the metric approximation property. \item[(ii)] The weak*-closure of the set of extreme points of the unit ball \({\mathcal B}_{{\mathcal H}^*}\) of the dual space \({\mathcal H}^*\), is the entire \({\mathcal B}_{{\mathcal H}^*}\). \end{itemize}} They also give a general method of construction of compact subsets \(X\) of \(U\) such that \({\mathcal F}(X)\) has a finite-dimensional decomposition and \({\mathcal F}(X)\) is not complemented in \({\mathcal H}\).
extreme points, finite dimensional decomposition
scientific article; zbMATH DE number 5306972
scientific article