# Definition:Vector Subspace/Hilbert Spaces

## 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 $\left({S, +, \circ}\right)_K$.

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 see

Compare with Definition:Closed Linear Subspace.

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*: $\S I.2$