# Vector space/Linear subspace/Closed/Definition

Linear subspace

Let be a
field,
and be a
-vector space.
A subset
is called a * linear subspace*, if the following properties hold.

- .
- If , then also .
- If and , then also holds.

Linear subspace

Let ${}K$ be a
field,
and ${}V$ be a
${}K$-vector space.
A subset
${}U\subseteq V$
is called a * linear subspace*, if the following properties hold.

- ${}0\in U$.
- If ${}u,v\in U$, then also ${}u+v\in U$.
- If ${}u\in U$ and ${}s\in K$, then also ${}su\in U$ holds.