Editing Glossary of tensor theory (section)

==Algebraic notation==
This avoids the initial use of components, and is distinguished by the explicit use of the tensor product symbol.

;Tensor product
:If ''v'' and ''w'' are vectors in [[vector space]]s ''V'' and ''W'' respectively, then
::<math>v \otimes w </math>
:is a tensor in
::<math>V \otimes W. </math>
:That is, the &otimes; operation is a [[binary operation]], but it takes values into a fresh space (it is in a strong sense ''external''). The &otimes; operation is a [[bilinear map]]; but no other conditions are applied to it.

;Pure tensor
:A pure tensor of ''V'' &otimes; ''W'' is one that is of the form ''v'' &otimes; ''w''.
:It could be written dyadically ''a<sup>i</sup>b<sup>j</sup>'', or more accurately ''a<sup>i</sup>b<sup>j</sup>'' '''e'''<sub>''i''</sub> &otimes; '''f'''<sub>''j''</sub>, where the '''e'''<sub>''i''</sub> are a basis for ''V'' and the '''f'''<sub>''j''</sub> a basis for ''W''. Therefore, unless ''V'' and ''W'' have the same dimension, the array of components need not be square. Such ''pure'' tensors are not generic: if both ''V'' and ''W'' have dimension greater than 1, there will be tensors that are not pure, and there will be non-linear conditions for a tensor to satisfy, to be pure. For more see [[Segre embedding]].

;Tensor algebra
:In the tensor algebra ''T''(''V'') of a vector space ''V'', the operation <math> \otimes </math> becomes a normal (internal) [[binary operation]]. A consequence is that ''T''(''V'') has infinite dimension unless ''V'' has dimension 0. The [[free algebra]] on a set ''X'' is for practical purposes the same as the tensor algebra on the vector space with ''X'' as basis.

;Hodge star operator

;Exterior power

:The [[wedge product]] is the anti-symmetric form of the &otimes; operation. The quotient space of ''T''(''V'') on which it becomes an internal operation is the ''[[exterior algebra]]'' of ''V''; it is a [[graded algebra]], with the graded piece of weight ''k'' being called the ''k''-th '''exterior power''' of ''V''.

;Symmetric power, symmetric algebra
:This is the invariant way of constructing [[polynomial algebra]]s.