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 $$\left({S, +, \circ}\right)_K$$.