# Definition Given an finite-[[Dimension|dimensional]] [[Inner Product Space]], we can associate with any [[vector]] $v$ in the [[vector space]] $V$ a [[one-form|dual vector]] $\tilde{v}$ in the [[Dual Space]] via the [[map]]: $ \begin{align} L: &V \rightarrow V^*, \\ &\pmb{v} \mapsto \tilde{v} \end{align} $ where $ L(\pmb{v}) = \braket{\pmb{v}|.} $ For example, $ \tilde{v}(\pmb{w}) = \braket{\pmb{v}|\pmb{w}} $ $\tilde{v}$ is known as the *metric dual* of $\pmb{v}$. Note that $L$ is an invertible [[map]]. It is also worth noting that the [[Dual Basis|dual basis]] $(e^i)$and the metric dual basis $L(e_i) = \tilde{e}^i$ are *not* the same in general. The former depends on the entire basis while the latter depends only on each vector.