Graded Vector Space - N-graded Vector Spaces

N-graded Vector Spaces

Let be the set of non-negative integers. An -graded vector space, often called simply a graded vector space without the prefix, is a vector space V which decomposes into a direct sum of the form

where each is a vector space. For a given n the elements of are then called homogeneous elements of degree n.

Graded vector spaces are common. For example the set of all polynomials in one variable form a graded vector space, where the homogeneous elements of degree n are exactly the linear combinations of monomials of degree n.