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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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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/

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[2] Olevskii A. M., “Raskhodyaschiesya ryady iz $L^2$ po polnym sistemam”, DAN SSSR, 138:3 (1961), 545–548 | MR | Zbl

[3] Price J. J., “Topics in orthogonal functions”, Amer. Math. Month., 82:6 (1975), 589–609 | DOI | MR

[4] Talalyan A. A., “O skhodimosti pochti vsyudu podposledovatelnostei chastnykh summ obschikh ortogonalnykh ryadov”, Izv. AN ArmSSR, ser. fiz.-matem. nauk, 10:3 (1957), 17–34 | MR

[5] Talalyan A. A., “Predstavlenie izmerimykh funktsii ryadami”, UMN, 15:5 (1960), 77–141 | MR | Zbl

[6] Nikishin E. M., Ulyanov P. L., “Ob absolyutnoi i bezuslovnoi skhodimosti”, UMN, 22:3 (1967), 240–242 | MR

[7] Olevskii A. M., Fourier series with respekt to general orthogonal sistems, Springer-Verlag, Berlin–Heidelberg–New York, 1975 | MR | Zbl

[8] Talalyan F. A., “Ob absolyutnoi i bezuslovnoi skhodimosti”, Izv. AN ArmSSR, ser. matem., 5:2 (1970), 109–137 | MR | Zbl