A tensor space of degree 
over a field 
is the vector space
tensor product 
of 
copies of a vector
space 
over 
. More generally, the tensor space of type 
is 
,
where 
is the dual vector space.
On a smooth manifold 
, the pointwise tensor space uses 
and 
. These spaces are the
fibers of a vector bundle.
For example,
 |
where 
is the tangent bundle and 
is the cotangent bundle.
A smooth tensor field is a bundle
section of this vector bundle. Its space of
global sections is formed using module
tensor products over the ring 
of smooth functions,
rather than vector space tensor products
over 
. The sections form
both a module over 
and a vector space
over 
.
