Separable Hilbert space, Riesz basis, Orthogonal basis, Analysis operator, Cofinite- rank operator


Necessary and sufficient conditions are obtained for a sequence $\{x_j:~j\in \mathbb J\}$ in a Hilbert space to be, up to the elimination of a finite subset of $\mathbb J$, the linear homeomorphic image of an orthogonal basis of some Hilbert space $K$. This extends a similar result for orthonormal bases due to Holub [J.R. Holub. Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces. \textit{Proc. Amer. Math. Soc.}, 122(3):779--785, 1994]. The proofs given here are based on simple linear algebra techniques.



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