Prove $\vec i^i = \vec i_i$ where $\vec i_i$ is a basis vector and $\vec i^i$ is a dual basis vector in rectangular cartesian coordinates.
My attempt:
$\vec i^i \cdot \vec i_j = \delta_j^i$
$\delta_j^i = 1$ if $i=j$
$\vec i^i \cdot \vec i_i = 1$
Since the bases in rectangular coordinate systems are orthonormal...
$\vec i_i = 1$
Given that $\vec i^i \cdot \vec i_i = 1$...
$\vec i_i = \vec i^i = 1$
Thanks!