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Let V be a n dimensional vector space, $\mu$ be an antisymmetric n tensor.(i.e, a real valued multilinear functional with n inputs) If there exists a basis for $V$, say, {$e_1,e_2,...,e_n$}, such that $\mu(e_1,...e_n)=1$,with corresponding dual basis $f_1,...f_n$ then how to show that $\mu=f^1\wedge f^2...\wedge f^n$?

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  • $\begingroup$ Considering the notation $\wedge$, is $\mu$ assumed to be symmetric or something? Otherwise, I do not think the result is true. $\endgroup$ – Najib Idrissi Oct 13 '14 at 18:27
  • $\begingroup$ @NajibIdrissi See my new edition $\endgroup$ – pxc3110 Oct 13 '14 at 18:33
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Up to a constant, there is only one such tensor, i.e. you know

$$ \mu=C f^1\wedge f^2...\wedge f^n $$

and just need to determine that $C = 1$; evaluating on the basis does precisely this.

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