# Graded vector space

In mathematics, a **graded vector space** is a vector space that has the extra structure of a *grading* or a *gradation*, which is a decomposition of the vector space into a direct sum of vector subspaces.

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

The subspaces of a graded vector space need not be indexed by the set of natural numbers, and may be indexed by the elements of any set *I*. An *I*-graded vector space *V* is a vector space together with a decomposition into a direct sum of subspaces indexed by elements *i* of set the *I*:

For general index sets *I*, a linear map between two *I*-graded vector spaces *f* : *V* → *W* is called a **graded linear map** if it preserves the grading of homogeneous elements. A graded linear map is also called a **homomorphism** (or **morphism**) of graded vector spaces, or **homogeneous linear map**:

For a fixed field and a fixed index set, the graded vector spaces form a category whose morphisms are the graded linear maps.

When *I* is a commutative monoid (such as the natural numbers), then one may more generally define linear maps that are **homogeneous** of any degree *i* in *I* by the property

where "+" denotes the monoid operation. If moreover *I* satisfies the cancellation property so that it can be embedded into a commutative group *A* that it generates (for instance the integers if *I* is the natural numbers), then one may also define linear maps that are homogeneous of degree *i* in *A* by the same property (but now "+" denotes the group operation in *A*). Specifically, for *i* in *I* a linear map will be homogeneous of degree −*i* if

Just as the set of linear maps from a vector space to itself forms an associative algebra (the algebra of endomorphisms of the vector space), the sets of homogeneous linear maps from a space to itself, either restricting degrees to *I* or allowing any degrees in the group *A*, form associative graded algebras over those index sets.

Some operations on vector spaces can be defined for graded vector spaces as well.

Given two *I*-graded vector spaces *V* and *W*, their **direct sum** has underlying vector space *V* ⊕ *W* with gradation