Dual Basis

Given a Contravariant Basis $\{\vec e_1,\ldots,\vec e_n\}$, its dual Covariant basis is given by

$$\vec e\,^\alpha \cdot \vec e_\beta = g(\vec e\,^\alpha, \vec e_\beta)= \delta^\alpha _\beta,$$

where $g$ is the Metric and $\delta^\alpha_\beta$ is the mixed Kronecker Delta. In Euclidean Space with an Orthonormal Basis,

$$\vec e\,^j = \vec e_j,$$

so the Basis and its dual are the same.