Tensors and Differential Forms

Remark. It is also possible to define tensors over a complex vector space (with no essential modifications in the definitions and results below, except that the special Euclidean case no longer applies). This is important in many other applications (including complex manifolds), but will not be considered here. 1.1. The dual space. Let V ∗ = hom(V ;R) be the dual of V , i.e. the vector space of all linear functionals V → R. We write 〈v∗, v〉 for v∗(v), where v∗ ∈ V ∗ and v ∈ V . Note that V and V ∗ are isomorphic vector spaces, since they have the same dimension, but there is, in general, no specific natural isomorphism between them and it is important to distinguish between them. (For the Euclidean case, see below.) On the other hand, there is a natural isomorphism V ∼= V ∗∗ between V and its second dual, and these space may be identified. (v ∈ V corresponds to v∗∗ ∈ V ∗∗ given by 〈v∗∗, v∗〉 = 〈v∗, v〉, v∗ ∈ V ∗.) If {e1, . . . , en} is a basis in V , so that the elements of V may uniquely be written ∑n 1 a ei, a i ∈ R, we define e, . . . , e ∈ V ∗ by