ALMOST-ORTHONORMAL BASES FOR HILBERT SPACE
Glasgow mathematical journal, Tome 47 (2005) no. 3, pp. 563-566
Voir la notice de l'article provenant de la source Cambridge
A basis $\{x_n\}$ for a Hilbert space H is called a Riesz basis if it has the property that $\sum a_nx_n$ converges in H if and only if $\sum|a_n|^2<\infty$, and hence if and only if $\{x_n\}$ is the isomorphic image of some orthonormal basis for H. A consequence of a classical result of Bary [1] is that any basis for H that is quadratically near an orthonormal basis must be a Riesz basis. Motivated by this result, we study in this paper the class of normalized bases in a Hilbert space that are quadratically near some orthonormal basis, bases we call almost-orthonormal bases. In particular, we prove that any such basis must be quadratically near its Gram-Schmidt orthonormalization, and derive an internal characterization of these bases that indicates how restrictive the property of being almost-orthonormal is.
HOLUB, JAMES R. ALMOST-ORTHONORMAL BASES FOR HILBERT SPACE. Glasgow mathematical journal, Tome 47 (2005) no. 3, pp. 563-566. doi: 10.1017/S0017089505002788
@article{10_1017_S0017089505002788,
author = {HOLUB, JAMES R.},
title = {ALMOST-ORTHONORMAL {BASES} {FOR} {HILBERT} {SPACE}},
journal = {Glasgow mathematical journal},
pages = {563--566},
year = {2005},
volume = {47},
number = {3},
doi = {10.1017/S0017089505002788},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089505002788/}
}
Cité par Sources :