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

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.

Compare Closed Linear Subspace.

Subspaces of $\R^n$
Let $\mathbb{W}\subset\R^n$.

$\mathbb{W}$ is a linear subspace of $\R^n$ iff:


 * $\mathbf{0} \in \mathbb{W}$, where $\mathbf{0}$ is the zero vector with $n$ entries, all $0$.


 * $\mathbb{W}$ is closed under vector addition


 * $\mathbb{W}$ is closed under scalar multiplication.

Some sources define the first condition as $\mathbb{W}$ being non-empty.

The two definitions are equivalent.

By Vector Subspace Test, it can be seen that these definitions are consistent with the definition given above for general vector subspaces.

Also see

 * Linear Subspaces Closed under Intersection
 * Linear Subspaces Closed under Setwise Addition
 * Submodule
 * Closed Linear Subspace