Editing Tensor contraction (section)

== Abstract formulation ==
Let ''V'' be a vector space over a [[field (mathematics)|field]] ''k''. The core of the contraction operation, and the simplest case, is the canonical pairing of ''V'' with its [[Dual space|dual vector space]] ''V''<sup>∗</sup>. The pairing is the [[linear map]] from the [[tensor product]] of these two spaces to the field ''k'':

: <math> C : V \otimes V^* \rightarrow k </math>

corresponding to the [[bilinear form]]

: <math> \langle v, f \rangle = f(v) </math>

where ''f'' is in ''V''<sup>∗</sup> and ''v'' is in ''V''. The map ''C'' defines the contraction operation on a tensor of type {{nowrap|(1, 1)}}, which is an element of <math>V \otimes V^* </math>. Note that the result is a [[scalar (mathematics)|scalar]] (an element of ''k''). In [[dimension of a vector space|finite dimension]]s, using the [[natural isomorphism]] between <math>V \otimes V^* </math> and the space of linear maps from ''V'' to ''V'',<ref name="natural iso">Let {{nowrap|L(''V'', ''V'')}} be the space of linear maps from ''V'' to ''V''. Then the natural map

: <math>V^* \otimes V \rightarrow L(V,V) </math>

is defined by

: <math>f \otimes v \mapsto g ,</math>

where {{nowrap|1=''g''(''w'') = ''f''(''w'')''v''}}. Suppose that ''V'' is finite-dimensional. If {''v''<sub>''i''</sub>} is a [[basis (vector space)|basis]] of ''V'' and {''f''<sup>''i''</sup>} is the corresponding dual basis, then <math>f^i \otimes v_j</math> maps to the transformation whose matrix in this basis has only one nonzero entry, a 1 in the ''i'',''j'' position. This shows that the map is an isomorphism.</ref> one obtains a basis-free definition of the [[trace (linear algebra)|trace]].

In general, a [[tensor]] of type {{nowrap|(''m'', ''n'')}} (with {{nowrap|''m'' ≥ 1}} and {{nowrap|''n'' ≥ 1}}) is an element of the vector space

: <math>V \otimes \cdots \otimes V \otimes V^{*} \otimes \cdots \otimes V^{*}</math>

(where there are ''m'' factors ''V'' and ''n'' factors ''V''<sup>∗</sup>).<ref name="fulton_harris">{{cite book |first1=William |last1=Fulton |author-link1=William Fulton (mathematician) |first2=Joe |last2=Harris |author-link2=Joe Harris (mathematician) |title=Representation Theory: A First Course |series=[[Graduate Texts in Mathematics|GTM]] |volume=129 |publisher=Springer |location=New York |year=1991 |isbn=0-387-97495-4 |pages=471–476 }}</ref><ref name="warner">{{cite book |first=Frank |last=Warner |title=Foundations of Differentiable Manifolds and Lie Groups |series=[[Graduate Texts in Mathematics|GTM]] |volume=94 |publisher=Springer |location=New York |year=1993 |isbn=0-387-90894-3 |pages=54–56 }}</ref> Applying the canonical pairing to the ''k''th ''V'' factor and the ''l''th ''V''<sup>∗</sup> factor, and using the identity on all other factors, defines the (''k'', ''l'') contraction operation, which is a linear map that yields a tensor of type {{nowrap|(''m'' − 1, ''n'' − 1)}}.<ref name="fulton_harris"/> By analogy with the {{nowrap|(1, 1)}} case, the general contraction operation is sometimes called the trace.