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

A vector subspace is also known as a linear subspace.

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

Caution
When considering Hilbert spaces, one wants to deal with projections onto subspaces.

These projections however require the linear subspace to be closed in topological sense in order to be well-defined.

In treatises of Hilbert spaces, one encounters the terminology linear manifold for the concept defined above.

A linear subspace is then also understood to be closed in topological sense.

Also see

 * Submodule