Let {fn}n=0∞ be a uniformly bounded sequence of real-valued measurable functions defined on [0,1] satisfying ∫01fn2=1. Further, let {cn} be a sequence of real numbers with n=0∑∞cn2=+∞. Prove that some re-arrangement of the series ∑n=0∞cnfn is divergent on a set of positive measure.
J. Komlos functionintegrationreal analysisreal analysis unsolved