Definition:Vector Subspace

Definition
Let $K$ be a division ring.

Let $\left({S, +, \circ}\right)_K$ be a $K$-algebraic structure with one operation.

Let $T$ be a closed subset of $S$.

Let $\left({T, +_T, \circ_T}\right)_K$ be an $K$-vector space where:
 * $+_T$ is the restriction of $+$ to $T \times T$ and
 * $\circ_T$ is the restriction of $\circ$ to $K \times T$.

Then $\left({T, +_T, \circ_T}\right)_K$ is a (vector) subspace of $\left({S, +, \circ}\right)_K$.

Also see

 * Submodule

Proper Subspace
If $T$ is a proper subset of $S$, then $\left({T, +_T, \circ_T}\right)_K$ is a proper (vector) subspace of $\left({S, +, \circ}\right)_K$.