Editing Tensor product (section)

=== Evaluation map and tensor contraction ===
For tensors of type {{math|(1, 1)}} there is a canonical '''evaluation map:'''
<math display="block">V \otimes V^* \to K</math>
defined by its action on pure tensors:
<math display="block">v \otimes f \mapsto f(v).</math>

More generally, for tensors of type {{tmath|1= (r, s) }}, with {{math|''r'', ''s'' > 0}}, there is a map, called [[tensor contraction]]:
<math display="block">T^r_s (V) \to T^{r-1}_{s-1}(V).</math>
(The copies of <math>V</math> and <math>V^*</math> on which this map is to be applied must be specified.)

On the other hand, if <math>V</math> is {{em|finite-dimensional}}, there is a canonical map in the other direction (called the '''coevaluation map'''):
<math display="block">\begin{cases} K \to V \otimes V^* \\ \lambda \mapsto \sum_i \lambda v_i \otimes v^*_i \end{cases}</math>
where <math>v_1, \ldots, v_n</math> is any basis of {{tmath|1= V }}, and <math>v_i^*</math> is its [[dual basis]]. This map does not depend on the choice of basis.<ref>{{Cite web| url= https://unapologetic.wordpress.com/2008/11/13/the-coevaluation-on-vector-spaces/|title=The Coevaluation on Vector Spaces|date=2008-11-13| website=The Unapologetic Mathematician|access-date=2017-01-26| url-status=live |archive-url =https://web.archive.org/web/20170202080439/https://unapologetic.wordpress.com/2008/11/13/the-coevaluation-on-vector-spaces/| archive-date =2017-02-02}}</ref>

The interplay of evaluation and coevaluation can be used to characterize finite-dimensional vector spaces without referring to bases.<ref>See [[Compact closed category]].</ref>