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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, Volume 25, Issue 2, Pages 244–247 (Mi vuu480)

MATHEMATICS

About one type of sequences that are not a Schauder basis in Hilbert spaces

A. Sh. Shukurov

Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, ul. B. Vahabzade, 9, Baku, AZ1141, Azerbaijan

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.

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UDC: 517.982
MSC: 46A35, 46B15, 46C05
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Citation: A. Sh. Shukurov, “About one type of sequences that are not a Schauder basis in Hilbert spaces”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 25:2 (2015), 244–247

Citation in format AMSBIB
\Bibitem{Shu15} \by A.~Sh.~Shukurov \paper About one type of sequences that are not a Schauder basis in Hilbert spaces \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2015 \vol 25 \issue 2 \pages 244--247 \mathnet{http://mi.mathnet.ru/vuu480} \elib{http://elibrary.ru/item.asp?id=23681105}