Subspaces

If $S$ is a nonempty subset of a vector space $V$, and $S$ satisfies the conditions:

1. $\alpha |x⟩\in S$ whenever $|x⟩\in S$ for any scalar $\alpha$
2. $|x⟩+|y⟩\in S$ whenever $|x⟩\in S$ and $|y⟩\in S$

then $S$ is said to be a subspace of $V$.