ϵ-Weak Cauchy sequences and a quantitative version of Rosenthal's ℓ1-theorem

A bounded sequence (xn) in a Banach space is called ϵ-weak Cauchy, for some ϵ>0, if for all x⁎∈BX⁎ there exists some n0∈N such that |x⁎(xn)−x⁎(xm)|<ϵ for all n≥n0 and m≥n0. It is shown that given ϵ>0 and a bounded sequence (xn) in a Banach space then either (xn) admits an ϵ-weak Cauchy subsequence or, for all δ>0, there exists a subsequence (xmn) with the following property. If I is a finite subset of N and ϕ:I→N∖I is any map then‖∑n∈Iλn(xmn−xmϕ(n))‖≥(ϵπ−δ)∑n∈I|λn| for every sequence of complex scalars (λn)n∈I. This provides an alternative proof for Rosenthal's ℓ1-theorem and strengthens its quantitative version due to Behrends. As a corollary we obtain that for any uniformly bounded sequence (fn) of complex-valued functions, continuous on the compact Hausdorff space K and satisfying lim⁡supn,m→∞|fn(t)−fm(t)|≤ϵ, for some ϵ>0 and all t∈K, there exists a subsequence (fjn) satisfying lim⁡supn,m→∞|∫K(fjn−fjm)dμ|≤2ϵ, for every Radon measure μ on K.