= Dual vector
{title2=$e^i$}
Dual vectors are the members of a <dual space>.
In the context of <tensors> , we use raised indices to refer to members of the dual basis vs the underlying basis:
$$
\begin{aligned}
e_1 & \in V \\
e_2 & \in V \\
e_3 & \in V \\
e^1 & \in V^* \\
e^2 & \in V^* \\
e^3 & \in V^* \\
\end{aligned}
$$
The dual basis vectors $e^i$ are defined to "pick the corresponding coordinate" out of elements of V. E.g.:
$$
\begin{aligned}
e^1 (4, -3, 6) & = 4 \\
e^2 (4, -3, 6) & = -3 \\
e^3 (4, -3, 6) & = 6 \\
\end{aligned}
$$
By expanding into the basis, we can put this more succinctly with the <Kronecker delta> as:
$$
e^i(e_j) = \delta_{ij}
$$
Note that in <Einstein notation>, the components of a dual vector have lower indices. This works well with the upper case indices of the dual vectors, allowing us to write a dual vector $f$ as:
$$
f = f_i e^i
$$
In the context of <quantum mechanics>, the <bra-ket>[bra] notation is also used for dual vectors.