The Haar system in L1 is monotonically boundedly complete

Анотація

Answering a question posed by J.R. Holub we show that for the normalized Haar system {h_n} in L₁[0, 1] whenever {a_n} is a sequence of scalars with |a_n| decreasing monotonically and with sup_N || Σ^N_n=1a_nh_n|| <∞, then Σ^∞_n=1a_nh_n converges in L₁[0, 1].