Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6417481 | Journal of Mathematical Analysis and Applications | 2016 | 6 Pages |
A bounded sequence (xn) in a Banach space is called ϵ-weak Cauchy, for some ϵ>0, if for all xââBXâ there exists some n0âN such that |xâ(xn)âxâ(xm)|<ϵ for all nâ¥n0 and mâ¥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 Ï:IâNâI is any map thenâânâIλn(xmnâxmÏ(n))ââ¥(ϵÏâδ)ânâI|λn| for every sequence of complex scalars (λn)nâI. This provides an alternative proof for Rosenthal's â1-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 limâ¡supn,mââ|fn(t)âfm(t)|â¤Ïµ, for some ϵ>0 and all tâK, there exists a subsequence (fjn) satisfying limâ¡supn,mââ|â«K(fjnâfjm)dμ|â¤2ϵ, for every Radon measure μ on K.