Let $X$ be a normed space (not necessarily complete) and $\{w_n\}_{n\in\mathbb N}\subset X$ be a linearly independent set. Let $\{x^*_n\}_{n\in\mathbb N}$ be a set of linear functionals on $X$ with the property that $x_i^*(w_j)=\delta_{ij}$. These linear functional are not necessarily bounded.

My question is the following: Is there any linear transformation (an analogue to Gramm-Schmidt procedure) that creates a new linearly independent set $\{u_n\}_{n\in\mathbb N}$, such that $$ \mathrm{span}\,\{u_n\}_{n\in\mathbb N}=\mathrm{span}\,\{w_n\}_{n\in\mathbb N} \subset X, $$ and for which the existence of a bounded dual set $\{u^*_n\}_{n\in\mathbb N}\subset X^*$, i.e., $u_i^*(u_j)=\delta_{ij}$, is guaranteed?