Differential Forms and Cohomology
نویسنده
چکیده
Definition 1. An m-linear function f which maps the m-fold cartesian product V m of a vector space V into some other vector space W is called alternating if f(v1, . . . , vm) = 0 whenever v1, . . . , vm ∈ V and vi = vj for i 6= j. We let ∧ (V ,W ) be the vector space of mlinear alternating functions mapping V m intoW . We then define ∧m V by the property that if f ∈ ∧m(V,W ) then there exists a unique h : ∧m V → W with f(v1, . . . , vm) = h(v1∧· · ·∧vm). Then associating f with h we obtain the linear isomorphism ∧m(V,W ) ∼= Hom(∧m V,W ). We will usually think of W as R. Proposition 1. A linear map f : V → V ′ induces a map ∧m(f,W ) : ∧m(V ′,W )→ ∧m(V,W ) defined by h ◦ f (m) V m f (m) −−→ (V ′)m h − → W
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