Characterizing Riesz bases via biorthogonal Bessel sequences

Abstract

Recently D.T. Stoeva proved that if two Bessel sequences in a separable Hilbert space $\mathcal H$ are biorthogonal and one of them is complete in $\mathcal H$, then both sequences are Riesz bases for $\mathcal H$. This improves a well known result where completeness is assumed on both sequences.

In this note we present an alternative proof of Stoeva's result which is quite short and elementary, based on the notion of Riesz-Fischer sequences.