Editing Tensor (intrinsic definition) (section)

==Definition via tensor products of vector spaces==
Given a finite set {{math|{{brace|''V''<sub>1</sub>, ..., ''V''<sub>''n''</sub>}}}} of [[vector space]]s over a common [[Field (mathematics)|field]] {{mvar|F}}, one may form their [[Tensor product#Tensor product of vector spaces|tensor product]] {{math|''V''<sub>1</sub> ⊗ ... ⊗ ''V''<sub>''n''</sub>}}, an element of which is termed a '''tensor'''.

A '''tensor on the vector space''' {{mvar|V}} is then defined to be an element of (i.e., a vector in) a vector space of the form:
<math display="block">V \otimes \cdots \otimes V \otimes V^* \otimes \cdots \otimes V^*</math>
where {{mvar|V{{sup|∗}}}} is the [[dual space]] of {{mvar|V}}.

If there are {{mvar|m}} copies of {{mvar|V}} and {{mvar|n}} copies of {{mvar|V{{sup|∗}}}} in our product, the tensor is said to be of {{nowrap|'''type ({{mvar|m}}, {{mvar|n}})'''}} and [[Covariance and contravariance of vectors|contravariant]] of order {{mvar|m}} and covariant of order {{mvar|n}} and of total [[tensor order|order]] {{math|''m'' + ''n''}}. The tensors of order zero are just the scalars (elements of the field {{mvar|F}}), those of contravariant order 1 are the vectors in {{mvar|V}}, and those of covariant order 1 are the [[linear functional|one-forms]] in {{mvar|V{{sup|∗}}}} (for this reason, the elements of the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type {{math|(''m'', ''n'')}} is denoted
<math display="block"> T^m_n(V) = \underbrace{ V\otimes \dots \otimes V}_{m} \otimes \underbrace{ V^*\otimes \dots \otimes V^*}_{n}.</math>

'''Example 1.''' The space of type {{math|(1, 1)}} tensors, <math>T^1_1(V) = V \otimes V^*,</math> is [[isomorphic]] in a natural way to the space of [[linear transformations]] from {{mvar|V}} to {{mvar|V}}. 

'''Example 2.''' A [[bilinear form]] on a real vector space {{mvar|V}}, <math>V\times V \to F,</math> corresponds in a natural way to a type {{math|(0, 2)}} tensor in <math>T^0_2 (V) = V^* \otimes V^*.</math> An example of such a bilinear form may be defined,{{clarify|date=October 2023}} termed the associated ''[[metric tensor]]'', and is usually denoted {{mvar|g}}.