On complete Riesz–Fischer sequences in a Hilbert space

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 $H$.