ALMOST-ORTHONORMAL BASES FOR HILBERT SPACE

Abstract

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.