information:extraction, quality:corruption, structural:summarization
As multilinear maps

A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object. Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred. One approach that is common in differential geometry is to define tensors relative to a fixed (finite-dimensional) vector space V, which is usually taken to be a particular vector space of some geometrical significance like the tangent space to a manifold. In this approach, a type (p, q) tensor T is defined as a multilinear map,

where V is the corresponding dual space of covectors, which is linear in each of its arguments. The above assumes V is a vector space over the real numbers, R {\displaystyle \mathbb {R} }. More generally, V can be taken over any field F (e.g. the complex numbers), with F replacing R {\displaystyle \mathbb {R} } as the codomain of the multilinear maps.

By applying a multilinear map T of type (p, q) to a basis { e } for V and a canonical cobasis { ε } for V,

a (p + q) -dimensional array of components can be obtained. A different choice of basis will yield different components. But, because T is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition. The multidimensional array of components of T thus form a tensor according to that definition. Moreover, such an array can be realized as the components of some multilinear map T. This motivates viewing multilinear maps as the intrinsic objects underlying tensors.

In viewing a tensor as a multilinear map, it is conventional to identify the double dual V of the vector space V, i.e., the space of linear functionals on the dual vector space V, with the vector space V. There is always a natural linear map from V to its double dual, given by evaluating a linear form in V against a vector in V. This linear mapping is an isomorphism in finite dimensions, and it is often then expedient to identify V with its double dual.