Dunford-Pettis property

AbstractThe following theorem is proved. If a locally convex space, quasi-complete for Mackey topology, has D-P (Dunford-Pettis) property then it has strict D-P property. Conversely, if (E′, σ(E′, E)) has a σ-compact dense subset and E has strict D-P property, then it has D-P property. Also it is proved that (Cb(X),F) where F=β0, β, or β1, has strict D-P property and (Cb(X), β0) has D-P property; if X contains a σ-compact dense subset then (Cb(X), β) and (Cb(X), β1) have D-P property