Let $\{ f_k \}_{k=1}^{\infty}$ be a Riesz basis with Riesz bounds $A$ and $B,$ and $\{ g_k \}_{k=1}^{\infty}$ be its dual Riesz basis. Show that $$\frac{1}{B} \leq \| g_k \|^2 \leq \frac{1}{A}.$$
My approach: Since $\{ f_k \}_{k=1}^{\infty}$ is a a Riesz basis the dual Riesz basis is $\{ S^{-1} f_k \}_{k=1}^{\infty},$ where $S:\mathcal{H} \rightarrow \mathcal{H}$ is a frame operator: $$Sf=TT^*f=\sum_{k=1}^{\infty} \left\langle f, f_k \right\rangle f_k .$$ Since $S$ is invertible and self-adjoint, we have $$\sum_{k=1}^{\infty} |\left\langle f, S^{-1}f_k \right\rangle|^2 = \sum_{k=1}^{\infty} |\left\langle S^{-1}f, f_k \right\rangle|^2 \leq B \|~ S^{-1}f \|^2 \leq B ~ \| S^{-1} \|^2 ~ \| f \|^2.$$
Further, we have $$\sum_{k=1}^{\infty} \left\langle f, S^{-1}f_k \right\rangle S^{-1}f_k = S^{-1} \sum_{k=1}^{\infty} \left\langle S^{-1}f, f_k \right\rangle f_k =S^{-1} S S^{-1} f =S^{-1}f.$$
From this, how to continue and get the desired inequality. Any help is much appreciated.
Hint: A Riesz basis $\{f_k\}_{k \in \mathbb{N}}$ with bounds $A,B > 0$ is, in particular, a frame with frame bounds $A,B$. The dual Riesz basis $\{S^{-1} f_k\}_{k \in \mathbb{N}}$ is the canonical dual frame of $\{f_k\}_{k \in \mathbb{N}}$, and hence has frame bounds $B^{-1}, A^{-1}$.