Let $\{ f_k \}_{k=1}^{\infty}$ be a Riesz basis with bounds $A$ and $B,$ and $\{ g_k \}_{k=1}^{\infty}$ be its dual Riesz basis. Show that $$\frac{1}{B} \leq \| g_k \| \leq \frac{1}{A}.$$
Note that $|<f_k,g_i>|=\delta_{k,i},$ where $\{ f_k \}_{k=1}^{\infty}$ and $\{ g_k \}_{k=1}^{\infty}$ are biorthogonal systems. Then we have $$1=|\langle f_k , g_k \rangle | \leq \| f_k \| \cdot \| g_k \| \leq B \cdot \| g_k \|,$$ which implies $$\| g_k \| \geq \frac{1}{B}.$$
I'm stuck in obtaining a corresponding upper bound. Any help is appreciated.
Note that your inequalies should be $$\frac{1}{B} \leq \| g_k \|^2 \leq \frac{1}{A},$$ i.e., the norm should be squared.
For the upper bound, suppose the system $\{ f_k \}_{k \in \mathbb{N}}$ is a Riesz basis with corresponding biorthogonal system $\{ g_k \}_{k \in \mathbb{N}}$. Then [1, Proposistion 3.7.3] implies that the system $\{g_k \}_{k \in \mathbb{N}}$ is a Bessel sequence with bound $A^{-1}$, i.e., for all $f \in \mathcal{H}$, it holds that $$ \sum_{k \in \mathbb{N}} | \langle f, g_k \rangle |^2 \leq \frac{1}{A} \|f \|_{\mathcal{H}}^2. $$ Take $g_j \in \{g_k \}_{k \in \mathbb{N}}$ with $j \in \mathbb{N}$. Then $g_j \in \mathcal{H}$, and $$ \| g_j \|_{\mathcal{H}}^4 = |\langle g_j , g_j \rangle|^2 \leq \sum_{k \in \mathbb{N}} | \langle g_j, g_k \rangle |^2 \leq \frac{1}{A} \|g_j \|_{\mathcal{H}}^2, $$ which yields the result.
[1] Christensen, O. An Introduction to Riesz Bases and Frames (second edition). Birkhauser, 2016.