A special class of Banach spaces
Theorem 1. Let \((e_ n)_ 1^{\infty}\) be a Schauder basis in a Banach space E, and \(\Gamma =[e^*_ n]_ 1^{\infty}\) its coefficient space. Then there exists an equivalent norm on E such that every non- separable subspace of \(E^*\) containing \(\Gamma\) does not have Auerbach bases. Theorem 2. There exists a Banach space which has an unconditional basis but does not have Auerbach bases. Theorem 3. Let E be a Banach space with a Schauder basis. Then there exists an equivalent norm on E in which every Auerbach basis is a 1- norming Markushevich basis. A biorthogonal system \((x_ i,x^*_ i)_{i\in I}\) is said to be an Auerbach basis if \((x_ i)_{i\in I}\) is a complete system and \(\| x_ i\| =\| x^*_ i\| =1\), \(i\in I\).
scientific article; zbMATH DE number 3916982