Differential Forms

5.1 Definition Let V be a vector space, and let V^k denote the cartesian product V × ⋯ × V with k factors. A multilinear map T: V^k → W into a vector space W is said to be skew symmetric if it changes sign whenever any pair of variables is interchanged, i.e., if https://www.w3.org/1998/Math/MathML"> T ( v 1 , … , v i , … , v j , … , v k ) = − T ( v 1 , … , v j , … , v i , … , v k ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315366722/660a41ec-09ea-4535-a2aa-9fc9a667124f/content/un5_e001.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for all v ₁, …,v_k ∈ V.