Editing Tensor (intrinsic definition) (section)

==Universal property==
The space <math>T^m_n(V)</math> can be characterized by a [[universal property]] in terms of [[multilinear map]]pings. Amongst the advantages of this approach are that it gives a way to show that many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis). Explicit computational information can then be written down using bases, and this order of priorities can be more convenient than proving a formula gives rise to a natural mapping. Another aspect is that tensor products are not used only for [[free module]]s, and the "universal" approach carries over more easily to more general situations.

A scalar-valued function on a [[Cartesian product]] (or [[Direct sum of modules|direct sum]]) of vector spaces
<math display="block">f : V_1\times\cdots\times V_N \to F</math>
is multilinear if it is linear in each argument. The space of all multilinear mappings from {{math|''V''<sub>1</sub> × ... × ''V<sub>N</sub>''}} to {{mvar|W}} is denoted {{math|''L<sup>N</sup>''(''V''<sub>1</sub>, ..., ''V<sub>N</sub>'';&nbsp;''W'')}}. When {{math|1= ''N'' = 1}}, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from {{mvar|V}} to {{mvar|W}} is denoted {{math|''L''(''V''; ''W'')}}.

The [[tensor product#Universal property|universal characterization of the tensor product]] implies that, for each multilinear function
<math display="block">f\in L^{m+n}(\underbrace{V^*,\ldots,V^*}_m,\underbrace{V,\ldots,V}_n;W)</math>
(where {{mvar|W}} can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function
<math display="block">T_f \in L(\underbrace{V^*\otimes\cdots\otimes V^*}_m \otimes \underbrace{V\otimes\cdots\otimes V}_n; W)</math>
such that
<math display="block">f(\alpha_1,\ldots,\alpha_m, v_1,\ldots,v_n) = T_f(\alpha_1\otimes\cdots\otimes\alpha_m \otimes v_1\otimes\cdots\otimes v_n)</math>
for all {{mvar|v{{sub|i}}}} in {{mvar|V}} and {{mvar|α{{sub|i}}}} in {{mvar|V{{sup|∗}}}}.

Using the universal property, it follows, when {{mvar|V}} is [[Dimension_(vector_space)|finite dimensional]], that the space of {{math|(''m'', ''n'')}}-tensors admits a [[natural isomorphism]]
<math display="block">T^m_n(V) \cong L(\underbrace{V^* \otimes \cdots \otimes V^*}_m \otimes \underbrace{V \otimes \cdots \otimes V}_n; F) \cong L^{m+n}(\underbrace{V^*, \ldots,V^*}_m,\underbrace{V,\ldots,V}_n; F).</math>

Each {{mvar|V}} in the definition of the tensor corresponds to a {{mvar|V{{sup|∗}}}} inside the argument of the linear maps, and vice versa. (Note that in the former case, there are {{mvar|m}} copies of {{mvar|V}} and {{mvar|n}} copies of {{mvar|V{{sup|∗}}}}, and in the latter case vice versa). In particular, one has
<math display="block">\begin{align}
T^1_0(V) &\cong L(V^*;F) \cong V,\\
T^0_1(V) &\cong L(V;F) = V^*,\\
T^1_1(V) &\cong L(V;V).
\end{align}</math>