Geodesic
On complete Riesz--Fischer sequences in a Hilbert space
Problemy analiza, Tome 13 (2024) no. 1, pp. 124-131.

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We prove that if $\{f_n\}_{n=1}^{\infty}$ is a complete Riesz–Fischer sequence in a separable Hilbert space $H$, then $$ T:=\{f\in H\colon \sum |\langle f, f_n\rangle |^2\infty\} $$ is closed in $H$ if and only if $\{f_n\}_{n=1}^{\infty}$ has a biorthogonal Riesz sequence. If the latter is also complete in $H$, then $\{f_n\}_{n=1}^{\infty}$ is a Riesz basis for $H$.
Keywords: Riesz–Fischer sequences, Bessel sequences, Riesz sequences, Riesz bases, completeness.
Mots-clés : biorthogonal sequences 
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E. Zikkos. On complete Riesz--Fischer sequences in a Hilbert space. Problemy analiza, Tome 13 (2024) no. 1, pp. 124-131. http://geodesic.mathdoc.fr/item/PA_2024_13_1_a7/

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