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Let X be a Banach space and {ej}j=1 be a sequence in X.The author shows e ∞that {ej}∞j+1 is a basic sequence if and only if ∑n∞=1 rnαnj converges for every j ＞ 1 and e ∞ - ∞∞ e∑∞n=1 rn ∑∞j=1 αnjej = ∑j=1(∑∞n=1 rnαnj)j holds for every choice of scalar vari