Geodesic
About one type of sequences that are not a Schauder basis in Hilbert spaces
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 2, pp. 244-247 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $H$ be a Hilbert space and a (not necessarily bounded) sequence of its elements $\{e_n\}_{n=1}^{\infty}$ has a bounded subsequence $\{e_{n_k}\}_{k=1}^{\infty}$ such that $|(e_{n_k},e_{n_m})| \geqslant \alpha > 0$ for all sufficiently large $k,m \in N, k \neq m$. It is proved that such a sequence $\{e_n\}_{n=1}^{\infty}$ is not a basic sequence and thus is not a Schauder basis in $H$. Note that the results of this paper generalize and offer a short and more simple proof of some recent results obtained in this direction.
Keywords: Schauder basis, basic sequence, Hilbert space, orthonormal sequence and orthonormal basis, weakly convergent sequences. 
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     title = {About one type of sequences that are not a {Schauder} basis in {Hilbert} spaces},
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A. Sh. Shukurov. About one type of sequences that are not a Schauder basis in Hilbert spaces. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 2, pp. 244-247. http://geodesic.mathdoc.fr/item/VUU_2015_25_2_a7/

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