One-Step Vector Subspace Test

Theorem
Let $V$ be a vector space over a division ring $K$.

Let $U \subseteq V$ be a non-empty subset of $V$ such that:


 * $\forall u, v \in U: \forall \lambda \in K: u + \lambda v \in U$

Then $U$ is a subspace of $V$.

Corollary
Let $U \subseteq V$ be a non-empty subset of $V$ such that:


 * $(1): \qquad \forall u \in U, \lambda \in K: \lambda u \in U$
 * $(2): \qquad \forall u, v \in U: u + v \in U$

Then $U$ is a subspace of $V$.

Proof of Corollary
Suppose that $(1)$ and $(2)$ hold.

From $(1)$, we obtain for every $\lambda \in K$ and $u \in U$ that $\lambda u \in U$.

An application of $(2)$ yields the condition of the main theorem.

Hence $U$ is a vector subspace of $V$.

Also see

 * Vector Subspace of Real Vector Space