Let V and W be vector spaces over K, let V >r := V × V × . . . × V (r-copies), and let φ : V> r → W be r-multilinear maps. Moreover, let

Sr denote the symmetric group on the numbers 1, . . . , r.

How to Show that π : V> r → W deﬁned by

π(v1 , . . . , vr ) := 1 /r!∑ sign(σ) φ(vσ(1) , . . . , vσ(r) )

, (the sumation over σ∈ Sr),is an alternating multilinear r-form without saying that φ is alternating?

Thank you