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

For "pure" vector spaces, the concept has been introduced in homological algebra, and it is widely used for graded algebras, which are graded vector spaces with additional structures.

Let ${\displaystyle \mathbb {N} }$ be the set of non-negative integers. An ${\textstyle \mathbb {N} }$-graded vector space, often called simply a graded vector space without the prefix ${\displaystyle \mathbb {N} }$, is a vector space V together with a decomposition into a direct sum of the form

where each ${\displaystyle V_{n}}$ is a vector space. For a given n the elements of ${\displaystyle V_{n}}$ are then called homogeneous elements of degree n.

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 the set I:

Therefore, an ${\displaystyle \mathbb {N} }$-graded vector space, as defined above, is just an I-graded vector space where the set I is ${\displaystyle \mathbb {N} }$ (the set of natural numbers).

The case where I is the ring ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ (the elements 0 and 1) is particularly important in physics. A ${\displaystyle (\mathbb {Z} /2\mathbb {Z} )}$-graded vector space is also known as a supervector space.