The Characterization Of Uniformly Eberlein Compact Sets In C₀（Γ）

In 1985,Argyros and Farmaki gave a topological characterization of uniformly Eberlein compact sets in c0(Γ),that is,a non-empty weakly compact set K?c0(Γ)is uniformly Eberlein compact if and only if for any ε>0,there is a decomposition Γ={Γm(ε):m∈N}and a sequence of natural numbers {k(m,ε):m ∈N} so that for ?x∈K and ?m∈N satisfying card{γ∈Γm(ε):|x(γ)|>ε}≤k(m,ε).Recently,Professor L.Cheng、Lancient and Raja have proved that the super weakly compact set is uniformly Eberlein compact by proving that the closed convex hull of a super weakly compact set is still a super weakly compact set.Motivated by the above two theorems,this thesis obtains the characterization of uniformly Eberlein compact sets in c0(Γ)by discussing in two cases through construction and Grothendiek-type theorem,that is,a non-empty weakly compact set K ?c0(Γ)is uniformly Eberlein compact if and only if K is a subset of the direct sum of countable pairwise disjoint super weakly compact sets.