Weak convergence in metric spaces and theory of concentrated compactness in Banach spaces

Abstract:  Delta-convergence is a mode of convergence in metric spaces, which is weaker than usual convergence. It coincides with weak convergence in Hilbert spaces, but not necessarily so in Banach spaces.  Delta-convergence appears in the fixed point theory, and, recently, in analysis of concentration: a bounded sequence in a Banach space is an asymptotic sum of blowups that are defined as deflating Delta-limits, rather than weak limits, of the sequence. In practice, harmonic analysis supplies Besov and TL spaces (including Sobolev spaces) with an equivalent norm for which Delta- and weak convergence coincide. This talk is dedicated to the memory of T.-C. Lim, who introduced Delta-convergence and proved a Delta-compactness theorem four decades ago.