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Azerbaijan Baku
Section  Mathematics 
Title  About one type of sequences that are not a Schauder basis in Hilbert spaces 
Author(s)  Shukurov A.Sh.^{a} 
Affiliations  Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences^{a} 
Abstract  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 
UDC  517.982 
MSC  46A35, 46B15, 46C05 
DOI  10.20537/vm150208 
Received  1 April 2015 
Language  English 
Citation  Shukurov A.Sh. About one type of sequences that are not a Schauder basis in Hilbert spaces, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 2, pp. 244247. 
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