On a characterization of Riesz bases via biorthogonal sequences
Diana T. Stoeva
TL;DR
This paper refines the characterization of Riesz bases in Hilbert spaces by showing that the completeness condition on one biorthogonal sequence can be removed, simplifying the criteria for Riesz bases.
Contribution
It proves that the completeness of either the original sequence or its biorthogonal sequence is unnecessary for the Riesz basis characterization.
Findings
Completeness of one biorthogonal sequence can be omitted
Simplifies the criteria for Riesz bases
Enhances understanding of biorthogonal sequences in Hilbert spaces
Abstract
It is well known that a sequence in a Hilbert space is a Riesz basis if and only if it is a complete Bessel sequence with biorthogonal sequence which is also a complete Bessel sequence. Here we prove that the completeness of one (any one) of the biorthogonal sequences can be omitted in the characterization.
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