Graded vector subspace

Let $𝑉$ be an $𝑆$ -graded vector space over $𝕂$ . A subspace $𝑊 \leq 𝑉$ is said to be graded iff $𝑊 = ⨁ 𝛼 \in 𝑆 𝑊 𝛼$ where $𝑊 𝛼 = 𝑊 \cap 𝑉 𝛼$ for $𝛼 \in 𝑆$ . linalg In this case one may construct the Quotient graded vector space so that it carries a natural $𝑆$ -gradation. See Graded submodule.