This may be a simple question, but I don't have any idea.
Consider $E$ to be $n$-dimensional Euclidean space with inner product $(\cdot,\cdot)$.
Let $\{v_1,\cdots,v_n\}$ be a basis of $E$ and let $\{w_1,\cdots,w_n\}$ be the dual basis w.r.t. $(\cdot,\cdot)$, i.e. $$(v_i,w_j)=\delta_{ij}.$$ Is there any way to express lengths of $w_i$'s in terms of lengths of $v_i$'s?
Length of $v\in E$ is as usual $\sqrt{(v,v)}.$
No, it's not possible.
On an exemple : $E=\mathbb{R^2}$
For a first basis $v_1 = (1,0)$ and $v_2 = ( 0,1)$ : both have length 1
The dual basis is $w_1 = (1,0)$ and $w_2 = ( 0,1)$ : both have lenght 1
Now consider a second basis $v_1' = (1,0)$ and $v_2' = ( \frac{\sqrt{2}}{2} ,\frac{\sqrt{2}}{2})$ both also have length 1
The dual basis is then $w_1 = (1,-\frac{\sqrt{2}}{2})$ and $w_2 = ( 0,\sqrt{2})$ : the length are not $1$
Conclusion : the length of the basis vectors are not enough information to get the lenghts of the dual basis vector