# 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**.

## Also see

- Set of Linear Subspaces is Closed under Intersection
- Linear Subspaces Closed under Setwise Addition
- Definition:Submodule
- Definition:Closed Linear Subspace
- Vector Subspace of Real Vector Space

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases - 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra - 1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur:
*Quantum Mechanics*... (previous) ... (next): Chapter $2$: Mathematical Background: $2.2$ Vector Spaces over $C$