Geodesic
On the connection between the properties of completeness and unconditional convergence for orthogonal systems
Sbornik. Mathematics, Tome 45 (1983) no. 4, pp. 507-513

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It is proved that there exists an orthonormal system that is not a system of unconditional convergence and, moreover, is not complete in the $L_2$-metric on any set of positive measure. Bibliography: 8 titles. 
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     author = {A. A. Talalyan},
     title = {On the connection between the properties of completeness and unconditional convergence for orthogonal systems},
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     url = {http://geodesic.mathdoc.fr/item/SM_1983_45_4_a6/}
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A. A. Talalyan. On the connection between the properties of completeness and unconditional convergence for orthogonal systems. Sbornik. Mathematics, Tome 45 (1983) no. 4, pp. 507-513. http://geodesic.mathdoc.fr/item/SM_1983_45_4_a6/