A characterization of the local dual spaces of a Banach space
In [Stud.\ Math.\ 147, No.\,2, 155--168 (2001; Zbl 0992.46007)], the authors of the paper under review defined the local dual spaces of a Banach space \(X\) as those closed subspaces \(Z\) of the dual space \(X^*\) such that for every pair of finite-dimensional subspaces \(E\) of \(X^*\) and \(F\) of \(X\), and every \(\varepsilon>0\), there exists an operator \(L:E \to Z\) which satisfies the following conditions: (a) \((1- \varepsilon) \| e\| \leq \| L(e)\| \leq (1+ \varepsilon) \| e\| \) for all \(e \in E\); (b) \(\langle L(e),x\rangle= \langle e,x\rangle\) for all \(e \in E\) and all \(x \in F\); (c) \(L(e)=e\) for all \(e \in E \cap Z\). Thus the classical principle of local reflexivity, as well as the principle of local reflexivity for ultrapowers, can be respectively stated as follows: (d) \(X\) is a local dual of \(X^*\); (e) for every ultrafilter \(\mathcal U\), \((X^{*})_{\mathcal U}\) is a local dual of the ultrapower \(X_{\mathcal U}\). In the present paper, the authors introduce the strict polar property as a test to check whether a closed subspace \(Z\) of \(X^*\) is a local dual of \(X\). As an application, they obtain a generalization of the principle of local reflexivity stated in (e). This new technical property also allows them to exhibit new examples of local dual spaces.
scientific article; zbMATH DE number 5083315
scientific article