Editing Tensor (section)

=== As multilinear maps ===
{{Main|Multilinear map}}
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.<ref>{{citation|last=Lee|first=John|title=Introduction to smooth manifolds|url={{google books |plainurl=y |id=4sGuQgAACAAJ|page=173}}|page=173|year=2000|publisher=Springer|isbn=978-0-387-95495-0}}</ref>  In this approach, a type {{nowrap|(''p'', ''q'')}} tensor ''T'' is defined as a [[multilinear map]],
:<math> T: \underbrace{V^* \times\dots\times V^*}_{p \text{ copies}} \times \underbrace{ V \times\dots\times V}_{q \text{ copies}} \rightarrow \mathbf{R}, </math>

where ''V''<sup>∗</sup> 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 number]]s, {{tmath|\R}}. More generally, ''V'' can be taken over any [[Field (mathematics)|field]] ''F'' (e.g. the [[complex number]]s), with ''F'' replacing {{tmath|\R}} as the codomain of the multilinear maps.

By applying a multilinear map ''T'' of type {{nowrap|(''p'', ''q'')}} to a basis {'''e'''<sub>''j''</sub>} for ''V'' and a canonical cobasis {'''ε'''<sup>''i''</sup>} for ''V''<sup>∗</sup>,
:<math>T^{i_1\dots i_p}_{j_1\dots j_q} \equiv T\left(\boldsymbol{\varepsilon}^{i_1}, \ldots,\boldsymbol{\varepsilon}^{i_p}, \mathbf{e}_{j_1}, \ldots, \mathbf{e}_{j_q}\right),</math>

a {{nowrap|(''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''<sup>∗∗</sup> of the vector space ''V'', i.e., the space of linear functionals on the dual  vector space ''V''<sup>∗</sup>, with the vector space ''V''.  There is always a [[Dual space#Injection into the double-dual|natural linear map]] from ''V'' to its double dual, given by evaluating a linear form in ''V''<sup>∗</sup> 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.