Definition:Vector Subspace

Definition

Let $K$ be a division ring.

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

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

Let $\struct {T, +_T, \circ_T}_K$ be a $K$-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 $\struct {T, +_T, \circ_T}_K$ is a (vector) subspace of $\struct {S, +, \circ}_K$.

Proper Subspace

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

Hilbert Spaces

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.

Therefore, in treatises of Hilbert spaces, one encounters the terminology linear manifold for the concept of vector subspace defined above.

The adapted definition of linear subspace is then that it is a topologically closed linear manifold.

Also known as

A vector subspace is also known as a linear subspace.