Linear Span is Linear Subspace/Proof 2

Theorem

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

Let $S \subseteq V$ be a subset of $V$.

Then the linear span $\map \span S$ is a subspace of $V$.

Proof

This is a special case of Generated Submodule is Linear Combinations.

As such, the statement follows immediately from that theorem.

$\blacksquare$