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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 Sciencesa
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. 244-247.
  1. Lyusternik L.A., Sobolev V.I. Elementy funktsional'nogo analiza (Elements of functional analysis), Moscow: Nauka, 1965.
  2. Young R.M. An introduction to nonharmonic Fourier series, New York: Academic Press, 1980.
  3. Mil'man V.D. Geometric theory of Banach spaces. Part I. The theory of basis and minimal systems, Russian Mathematical Surveys, 1970, vol. 25, issue 3, pp. 111-170.
  4. Khmyleva T.E., Bukhtina I.P. On some sequence of Hilbert space elements, which is not basis, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2007, no. 1, pp. 58-62 (in Russian).
  5. Khmyleva T.E., Ivanova O.G. On some systems of a Hilbert space which are not bases, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2010, no. 3 (11), pp. 53-60 (in Russian).
  6. Sadybekov M.A., Sarsenbi A.M. On a necessary condition for a system of normalized elements to be a basis in a Hilbert space, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2011, no. 1 (13), pp. 44-46 (in Russian).
  7. Dunford N., Schwartz J. Lineinye operatopy. Obshchaya teoriya (Linear operators. General theory), Moscow: Izd. Inostr. Lit., 1962, 895 p.
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