+7 (3412) 91 60 92

## Archive of Issues

Azerbaijan Baku
Year
2015
Volume
25
Issue
2
Pages
244-247
 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. References Lyusternik L.A., Sobolev V.I. Elementy funktsional'nogo analiza (Elements of functional analysis), Moscow: Nauka, 1965. Young R.M. An introduction to nonharmonic Fourier series, New York: Academic Press, 1980. 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. 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). 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). 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). Dunford N., Schwartz J. Lineinye operatopy. Obshchaya teoriya (Linear operators. General theory), Moscow: Izd. Inostr. Lit., 1962, 895 p. Full text