Efimov-Stechkin spaces (Q1098052)

Let X be a Banach space, \(X^*\) its dual and S, \(S^*\) the unit spheres in these spaces. X is said to be Efimov-Stecklin's space (ES-space) if for each \(x_ n\in S\) and \(x^*\in S^*\) such that \(x^*(x_ n)\to 1\) there exists a convergent subsequence \(\{x_{n_ k}\}\). The paper gives a survey of the properties of ES-spaces. Theorem 1 contains various definitions of ES-space, Theorem 2,3 characterize these spaces by means of their approximative properties; Theorem 4 concerns the solution of non-stable problem in ES-space.