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 $L_1[0,1]$ whenever $\{a_n\}$ is a sequence of scalars with $|a_n|$ decreasing monotonically and with $\sup_N || \sum_{n=1}^{\mathbb{N}}a_nh_n||<\infty$, then$ \sum_{n=1}^{\infty}a_nh_n$ converges in $L_1[0,1]$.