The question is of general interest for dimensional analysis. I am trying to formalize the dimensionality of vector spaces.
Consider the usual orthonormal basis of the Euclidean space $e_1, e_2, e_3$. If we define the reciprocal basis as $$e^i = e_i / ( e_i \cdot e_i)$$ does it mean that the physical dimension of $e^i$ is also reciprocal to the dimension of $e_1$? If $[e_1] =L $, length, is $[e^1]=1/L$?
Working with components, $e^i_j=e_{ij}/\sum_ke_{ik}^2$ is a length over a squared length. Since each component of $e^i$ has dimension $L^{-1}$, so does the vector.