Definition:Vector Subspace

From ProofWiki
Jump to navigation Jump to search

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.


Examples

Real Numbers in Complex Vector Space

The set of real numbers $\R$ forms a vector subspace of the vector space of complex numbers $\C$.


Real Numbers in Quaternion Vector Space

The set of complex numbers $\C$ forms a vector subspace of the vector space of quaternions $\H$.


Also known as

A vector subspace is also known as a linear subspace.

A vector subspace is frequently referred just as a subspace if it has been established what it is a subspace of.


Also see

  • Results about vector subspaces can be found here.

Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Vector_Subspace&oldid=726283#Proper_Subspace"