Geodesic
The Haar system in $L_1$ is monotonically boundedly complete
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 12 (2005) no. 1, pp. 103-106 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $L_1[0,1]$. 
@article{JMAG_2005_12_1_a5,
     author = {Vladimir Kadets},
     title = {The {Haar} system in $L_1$ is monotonically boundedly complete},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {103--106},
     year = {2005},
     volume = {12},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2005_12_1_a5/}
}
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Vladimir Kadets. The Haar system in $L_1$ is monotonically boundedly complete. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 12 (2005) no. 1, pp. 103-106. http://geodesic.mathdoc.fr/item/JMAG_2005_12_1_a5/