Let $V$ be a finite-dimensional [vector space](/page/Vector%20Space) over a field $k$, and let
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\begin{align*}
\iota_V: V \to V^{**}
\end{align*}
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be the canonical bidual map defined by $\iota_V(v)(\varphi)=\varphi(v)$ for every $v\in V$ and every $\varphi\in V^*$. Then $\iota_V$ is a $k$-linear isomorphism.
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Moreover, if $\mathcal B=(e_1,\ldots,e_n)$ is an ordered basis of $V$ and $\mathcal B^*=(e_1^*,\ldots,e_n^*)$ is its [dual basis](/theorems/414) in $V^*$, then the dual basis of $\mathcal B^*$ in $V^{**}$ is