Locally convex spaces not containing \(l^1\) (Q740790)

Banach spaces without isomorphic copies of \(\ell^1\) are characterized in Rosenthal's celebrated theorem by the condition that every bounded sequence has a weak Cauchy subsequence. Some more characterizations were proved by Dor and Mayoral. The present article gives versions of these characterizations for locally complete locally convex spaces with metrizable bounded sets. This class contains all Fréchet spaces, most of their duals (namely, locally complete (DF)-spaces satisfying Heinrich's dual density condition), as well as retractive (LF)-spaces.