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''of valency  $  s \geq  1 $''

A tensor of type  $  ( 0, s) $, 
an element of the tensor product  $  T _ {s} ( E) = E  ^ {*} \otimes \dots \otimes E  ^ {*} $
of  $  s $
copies of the dual space  $  E  ^ {*} $
of the vector space  $  E $
over a field  $  K $. 
The space  $  T _ {s} ( E) $
is itself a vector space over  $  K $
with respect to the addition of covariant tensors of the same valency and multiplication of them by scalars. Let  $  E $
be finite dimensional, let  $  e _ {1} \dots e _ {n} $
be a basis of  $  E $
and let  $  e  ^ {1} \dots e  ^ {n} $
be the basis dual to it of  $  E  ^ {*} $. 
Then  $   \mathop{\rm dim}  T _ {s} ( E) = n  ^ {s} $
and the set of all tensors of the form  $  e ^ {i _ {1} } \otimes \dots \otimes e ^ {i _ {s} } $, 
$  1 \leq  i _ {1} \dots i _ {s} \leq  n $, 
forms a basis for  $  T _ {s} ( E) $. 
Any covariant tensor can be represented in the form  $  t = t _ {i _ {1}  \dots i _ {s} } e ^ {i _ {1} } \otimes \dots \otimes e ^ {i _ {s} } $. 
The numbers  $  t _ {i _ {1}  \dots i _ {s} } $
are called the coordinates, or components, of the covariant tensor relative to the basis  $  e _ {1} \dots e _ {n} $
of  $  E $. 
Under a change of a basis of  $  E $
according to the formulas  $  e _ {j}  ^  \prime  = a _ {j}  ^ {i} e _ {i} $
and the corresponding change of the basis of  $  T _ {s} ( E) $, 
the components of the covariant tensor  $  t $
are changed according to the so-called covariant law

$$ 
t _ {j _ {1}  \dots j _ {s} }  ^  \prime   = \ 
a _ {j _ {1}  } ^ {i _ {1} } \dots
a _ {j _ {s}  } ^ {i _ {s} }
t _ {i _ {1}  \dots i _ {s} } .
$$

If  $  s = 1 $, 
the covariant tensor is called a [[Covariant vector|covariant vector]]; when  $  s \geq  2 $
a covariant tensor corresponds in an invariant way with an  $  s $-
linear mapping from the direct product  $  E  ^ {s} = E \times \dots \times E $(
$  s $
times) into  $  K $
by taking the components of the covariant tensor  $  t $
relative to the basis  $  e _ {1} \dots e _ {n} $
as the values of the  $  r $-
linear mapping  $  \widetilde{t}  $
at the basis vectors  $  ( e _ {i _ {1}  } \dots e _ {i _ {s}  } ) $
in  $  E  ^ {s} $, 
and conversely; for this reason a covariant tensor is sometimes defined as a multilinear functional on  $  E  ^ {s} $.

For references see [[Covariant vector|Covariant vector]].