Editing Tensor (section)

=== Using tensor products ===
{{Main|Tensor (intrinsic definition)}}
For some mathematical applications, a more abstract approach is sometimes useful.  This can be achieved by defining tensors in terms of elements of [[tensor product]]s of vector spaces, which in turn are defined through a [[universal property]] as explained [[Tensor product#Universal property|here]] and [[Tensor (intrinsic definition)#Universal property|here]].

A '''type {{math|(''p'', ''q'')}} tensor''' is defined in this context as an element of the tensor product of vector spaces,<ref>{{cite book|last1=Dodson|first1=C.T.J.|title=Tensor geometry: The Geometric Viewpoint and Its Uses |edition=2nd |date=2013 |isbn=9783642105142 |volume=130|orig-year=1991|series=Graduate Texts in Mathematics |publisher=Springer|last2=Poston |first2=T. |page= 105}}</ref><ref>{{Springer|id=a/a011120|title=Affine tensor}}</ref>
:<math>T \in \underbrace{V \otimes\dots\otimes V}_{p\text{ copies}} \otimes \underbrace{V^* \otimes\dots\otimes V^*}_{q \text{ copies}}.</math>

A basis {{math|''v''<sub>''i''</sub>}} of {{math|''V''}} and basis {{math|''w''<sub>''j''</sub>}} of {{math|''W''}} naturally induce a basis {{math|''v''<sub>''i''</sub> ⊗ ''w''<sub>''j''</sub>}} of the tensor product {{math|''V'' ⊗ ''W''}}.  The components of a tensor {{math|''T''}} are the coefficients of the tensor with respect to the basis obtained from a basis {{math|<nowiki>{</nowiki>'''e'''<sub>''i''</sub><nowiki>}</nowiki>}} for {{math|''V''}} and its dual basis {{math|{'''''ε'''''{{i sup|''j''}}<nowiki>}</nowiki>}}, i.e.
:<math>T = T^{i_1\dots i_p}_{j_1\dots j_q}\; \mathbf{e}_{i_1}\otimes\cdots\otimes \mathbf{e}_{i_p}\otimes \boldsymbol{\varepsilon}^{j_1}\otimes\cdots\otimes \boldsymbol{\varepsilon}^{j_q}.</math>

Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type {{math|(''p'', ''q'')}} tensor.  Moreover, the universal property of the tensor product gives a [[bijection|one-to-one correspondence]] between tensors defined in this way and tensors defined as multilinear maps.

This 1 to 1 correspondence can be achieved in the following way, because in the finite-dimensional case there exists a canonical isomorphism between a vector space and its double dual:

:<math>U \otimes V \cong\left(U^{* *}\right) \otimes\left(V^{* *}\right) \cong\left(U^{*} \otimes V^{*}\right)^{*} \cong \operatorname{Hom}^{2}\left(U^{*} \times V^{*} ; \mathbb{F}\right)</math>

The last line is using the universal property of the tensor product, that there is a 1 to 1 correspondence between maps from <math>\operatorname{Hom}^{2}\left(U^{*} \times V^{*} ; \mathbb{F}\right)</math> and <math>\operatorname{Hom}\left(U^{*} \otimes V^{*} ; \mathbb{F}\right)</math>.<ref>{{Cite web |date=June 5, 2021|title=Why are Tensors (Vectors of the form a⊗b...⊗z) multilinear maps? |url=https://math.stackexchange.com/q/4163471 |website=Mathematics Stackexchange}}</ref>

Tensor products can be defined in great generality&nbsp;– for example, [[tensor product of modules|involving arbitrary modules]] over a ring.  In principle, one could define a "tensor" simply to be an element of any tensor product.  However, the mathematics literature usually reserves the term ''tensor'' for an element of a tensor product of any number of copies of a single vector space {{math|''V''}} and its dual, as above.