Juan Carlos Ferrando (Operations Research Center, University Miguel Hernández of Elche) and Salvador López-Alfonso (Depto. Construcciones Arquitectónicas, Universitat Politècnica de València)
Abstract: A subset A of a locally convex space E is called (relatively) sequentially complete if every Cauchy sequence {xn}n=1∞ in E contained in A converges to a point x∈ A (a point x∈ E). Asanov and Velichko proved that if X is countably compact, every functionally bounded set in Cp(X) is relatively compact, and Baturov showed that if X is a Lindelöf Σ -space, each countably compact (so functionally bounded) set in Cp(X) is a monolithic compact. We show that if X is a Lindelöf Σ -space, every functionally bounded (relatively) sequentially complete set in Cp(X) or in Cw(X) , i. e., in Ck(X) equipped with the weak topology, is (relatively) Gul’ko compact. We get some consequences.