nLab linear subspace

Idea

In linear algebra, by linear subspaces are typically meant the subobjects of vector spaces, hence the sub-vector-spaces:

For $k$ a ground field and $V \in k Vec$ a $k$-vector space, then a linear subspace of $V$ is another $k$-vector space $W$ equipped with a monomorphism $i \colon W \hookrightarrow V$, hence with a linear map $i$ whose underlying map is injective.

Equivalently this means that the underlying set of $W$ is a subset of the underlying set of $V$ which is closed under the vector space operations in $V$.

If $V$ is equipped with the structure of a topological vector space then one is often interested in the closed linear subspaces.

More generally, for $R$ any ground ring one may regard $R$-modules as “$R$-linear spaces”. Understood in this generality, linear subspaces are the subobjects in the category of modules, hence the submodules.

Related concepts

• linear map

• subobject, subspace

• Hilbert lattice, quantum logic

Literature

See also:

• Wikipedia, Linear subspace