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.

Let ℕ be the set of non-negative integers. An ℕ-graded vector space, often called simply a graded vector space without the prefix ℕ, is a vector space V which decomposes 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 that can be written as a direct sum of subspaces indexed by elements i of 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).

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