Definition:Vector Subspace

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

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

Compare submodule.