LetXbeaBanachspaceand{ej}j=1beasequenceinX.Theauthorshowse∞that{ej}∞j+1isabasicsequenceifandonlyif∑n∞=1rnαnjconvergesforeveryj＞1ande∞-∞∞e∑∞n=1rn∑∞j=1αnjej=∑j=1(∑∞n=1rnαnj)jholdsforeverychoiceofscalarvariables{αnj}suchthat∑∞n=1αnjejconvergesforeachn>-1andanychoiceofscalarvariables{rn}suchthat∑∞n+1∑∞f=1rnαnjejconverges.Moreover,someapplicationsabouttheresultaregiven.
