Multilinear form

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a> and <a href="/facts/Multilinear_algebra/LrwpUFsM">multilinear algebra</a>, a multilinear form on a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
 
 
 
 V
 
 
 {\displaystyle V}
 
 over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is a <a href="/facts/Map_(mathematics)/FABF1l2f">map</a>

f
        :
        
          V
          
            k
          
        
        →
        K
      
    
    {\displaystyle f\colon V^{k}\to K}

that is separately 
 
 
 
 K
 
 
 {\displaystyle K}
 
-<a href="/facts/Linear/fChtg3KC">linear</a> in each of its 
 
 
 
 k
 
 
 {\displaystyle k}
 
 arguments. More generally, one can define multilinear forms on a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> over a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>. The rest of this article, however, will only consider multilinear forms on <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite-dimensional</a> vector spaces.
A multilinear 
 
 
 
 k
 
 
 {\displaystyle k}
 
-form on 
 
 
 
 V
 
 
 {\displaystyle V}
 
 over 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 is called a (covariant) 
 
 
 
 
 k
 
 
 
 {\displaystyle {\boldsymbol {k}}}
 
-tensor, and the vector space of such forms is usually denoted 
 
 
 
 
 
 
 T
 
 
 
 k
 
 
 (
 V
 )
 
 
 {\displaystyle {\mathcal {T}}^{k}(V)}
 
 or 
 
 
 
 
 
 
 L
 
 
 
 k
 
 
 (
 V
 )
 
 
 {\displaystyle {\mathcal {L}}^{k}(V)}
 
.

Multilinear form open-in-new

Multilinear form