Tensor (intrinsic Definition) - Definition Via Tensor Products of Vector Spaces

Definition Via Tensor Products of Vector Spaces

Given a finite set { V₁, ..., V_n } of vector spaces over a common field F, one may form their tensor product V₁ ⊗ ... ⊗ V_n, an element of which is termed a tensor.

A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form:

where V* is the dual space of V.

If there are m copies of V and n copies of V* in our product, the tensor is said to be of type (m, n) and contravariant of order m and covariant order n and total order m+n. The tensors of order zero are just the scalars (elements of the field F), those of contravariant order 1 are the vectors in V, and those of covariant order 1 are the one-forms in V* (for this reason the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (m,n) is denoted

The (1,1) tensors

are isomorphic in a natural way to the space of linear transformations from V to V. A bilinear form on a real vector space V; V × V → R corresponds in a natural way to a (0,2) tensor in

termed the associated metric tensor (or sometimes misleadingly the metric or inner product) and usually denoted g.