Norms that locally depend on countably many linear functionals (Q2766073)

The paper is devoted to study properties of Banach spaces \(X\), which have a norm that locally depends on countably many linear functionals, i.e., for every \(x\in X\) there is a neighborhood \(U\) of \(x\), countably many functionals \(\left( f_{i}\right) \subset X^{\ast}\) of unit norm and a continuous function \(\psi\) on \(\ell_{\infty}\) such that \(\left\|z\right\|=\psi\left( f_{1}\left( z\right) ,f_{2}\left( z\right) ,...\right) \) for each \(z\in U\). A typical result: if a Banach space \(X\) has such a norm, then it is either isomorphic to a subspace of \(\ell_{\infty}\) or contains an isomorphic copy of \(c_{0}.\) It is shown that such norms are closely related to the countable tightness of the weak* topology of dual spaces, to the projection resolution of identity, to Valdivia compacts, to biorthogonal systems, and so on. Besides, the paper contains a lot of examples and some open problems.