# the Exterior Power of Dual Spaces

We have known that the exterior power of vector spaces exists. We give a more convenient construction for finite dual spaces.

Let $\tilde \pi$ be filled in the commutative diagrams:
\begin{xy}
\xymatrix{
\Pi^n V^* \ar[r]^\wedge \ar[rd]_\pi & \wedge^n V^* \ar@{–>}[d]^{\tilde\pi}\\
& (\Pi^n V)^*,
}
\end{xy}
where $\pi$ is an alternating operator defined by $$\pi(v’_1,\cdots,v’_n):(v_1,\cdots,v_n)\mapsto \det(v’_i(v_j)).$$
We will show $\tilde\pi$ is an injection.