Topological characterisation of weakly compact operators

Let X be a Banach space. Then there is a locally convex topology for X, the “Right topology,” such that a linear map T , from X into a Banach space Y , is weakly compact, precisely when T is a continuous map from X, equipped with the “Right” topology, into Y equipped with the norm topology. When T is only sequentially continuous with respect to the Right topology, it is said to be pseudo weakly compact. This notion is related to Pelczynski’s Property (V ). © 2006 Elsevier Inc. All rights reserved.