Geodesic
About one type of sequences that are not a Schauder basis in Hilbert spaces
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 2, pp. 244-247

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{VUU_2015_25_2_a7,
     author = {A. Sh. Shukurov},
     title = {About one type of sequences that are not a {Schauder} basis in {Hilbert} spaces},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {244--247},
     publisher = {mathdoc},
     volume = {25},
     number = {2},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VUU_2015_25_2_a7/}
}
TY  - JOUR
AU  - A. Sh. Shukurov
TI  - About one type of sequences that are not a Schauder basis in Hilbert spaces
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2015
SP  - 244
EP  - 247
VL  - 25
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VUU_2015_25_2_a7/
LA  - en
ID  - VUU_2015_25_2_a7
ER  - 
%0 Journal Article
%A A. Sh. Shukurov
%T About one type of sequences that are not a Schauder basis in Hilbert spaces
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2015
%P 244-247
%V 25
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VUU_2015_25_2_a7/
%G en
%F VUU_2015_25_2_a7
A. Sh. Shukurov. About one type of sequences that are not a Schauder basis in Hilbert spaces. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 2, pp. 244-247. http://geodesic.mathdoc.fr/item/VUU_2015_25_2_a7/