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Section
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Mathematics
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Title
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About one type of sequences that are not a Schauder basis in Hilbert spaces
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Author(-s)
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Shukurov A.Sh.a
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Affiliations
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Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciencesa
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Abstract
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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.
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Keywords
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Schauder basis, basic sequence, Hilbert space, orthonormal sequence and orthonormal basis, weakly convergent sequences
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UDC
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517.982
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MSC
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46A35, 46B15, 46C05
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DOI
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10.20537/vm150208
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Received
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1 April 2015
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Language
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English
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Citation
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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. 244-247.
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References
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