Quasi-Suslin weak duals

Cascales, K¸akol, and Saxon (CKS) ushered Kaplansky and Valdivia into the grand setting of Cascales/Orihuela spaces E by proving: (K) If E is countably tight, then so is the weak space (E, σ(E,E )), and (V) (E, σ(E,E )) is countably tight iff weak dual (E ,σ(E ,E)) is K-analytic. The ensuing flow of quasi-Suslin weak duals that are not K-analytic, a la Valdivia’s example, continues here, where we argue that locally convex spaces E with quasi-Suslin weak duals are (K, V)’s best setting: largest by far, optimal vis-a-vis Valdivia. The vaunted CKS setting proves not larger, in fact, than Kaplansky’s. We refine and exploit the quasi-LB strong dual interplay. © 2007 Elsevier Inc. All rights reserved