Cyril Tintarev: Definitions of weak convergence in Banach spaces - a lecture in memory of Teck-Cheong Lim

A sequence in a metric space is called Delta-convergent (the notion introduced by T.C. Lim in 1976) to a point x, if x is an asymptotic center of its every subsequence. In Hilbert spaces Delta-convergence coincides with weak convergence. A metric space is called asymptotically complete if every bounded sequence in it has an asymptotic center. Lim's Delta-compactness theorem says that every bounded sequence in an asymptotically complete metric space has a Delta-convergent subsequence. This is a remarkable "fork" of Banach's notion of weak convergence, which in general Banach spaces is distinct from weak convergence. We discuss heuristic connection between asymptotic completeness and reflexivity in Banach spaces and applications of Delta-compactness to the fixed point theory, energy decoupling inequalities (Brezis-Lieb lemma), and defect of compactness in Banach spaces. The talk is based on a survey by Devillanova, Solimini, and the speaker, to be published by the AMS.