Editing Graded vector space (section)

== Integer gradation ==
Let <math>\mathbb{N}</math> be the set of non-negative [[integer]]s. An <math display="inline">\mathbb{N}</math>'''-graded vector space''', often called simply a '''graded vector space''' without the prefix <math>\mathbb{N}</math>, is a vector space {{math|''V''}} together with a decomposition into a direct sum of the form

: <math>V = \bigoplus_{n \in \mathbb{N}} V_n</math>
where each <math>V_n</math> is a vector space.  For a given ''n'' the elements of <math>V_n</math> are then called '''homogeneous''' elements of degree ''n''.

Graded vector spaces are common.  For example the set of all [[polynomial]]s in one or several variables forms a graded vector space, where the homogeneous elements of degree ''n'' are exactly the linear combinations of [[monomial]]s of [[degree of a polynomial|degree]]&nbsp;''n''.