Two-norm convergence and Tauberian boundedness theorems (Q1110752)

Two-norm convergence has turned out to be a very valuable tool in functional analysis and its applications. For example, two-norm convergence in the proper or generalised sense combines a boundedness property and a limit concept [see \textit{J. B. Cooper}, Saks spaces and applications to functional analysis, (1978; Zbl 0379.46005)] and \textit{A. Wilansky} [Summability through functional analysis. (1984; Zbl 0531.40008)]. The authors first introduce B-perfect and BM-perfect methods which are defined by using a two-norm convergence. Using these notations the authors investigate interesting relations between Tauberian boundedness conditions and Tauberian convergence conditions.