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Short Notes

The Haar system in $L_1$ is monotonically boundedly complete

Vladimir Kadets

Department of Mechanics and Mathematics, V.N. Karazin Kharkov National University, 4 Svobody Sq., Kharkov, 61077, Ukraine

Abstract: 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 $\sup_N\|\sum_{n=1}^N a_n h_n\| < \infty$, then $ \sum_{n=1}^\infty a_n h_n$ converges in $L_1[0,1]$.

Key words and phrases: Haar system; martingale; monotonically boundedly complete basis.

MSC: 46B15, 60G46

Received: 13.08.2004

Language: English

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