Let E be a linear space over a field K. Then the vector space tensor product ⊗_(λ = 1)^k E is called a tensor space of degree k. More specifically, a tensor space of type (r, s) can be described as a vector space tensor product between r copies of vector fields and s copies of the dual vector fields, i.e., one-forms. For example, T^(3, 1) = T M⊗T M⊗T M⊗T^* M is the vector bundle of (3, 1) tensors on a manifold M. Tensors of type (r, s) form a vector space.

tensor | vector space