Let $(V, F, \cdot)$ be a vector space. $(W, F, \cdot)$ is a vector subspace of $V$ if it is a vector space and $W \subseteq V$.
Let $V$ be a vector space over $F$ and $W \subseteq V$. $W$ is a subspace as soon as:
- \[0 \in W.\]
(Closure under addition) For all $x$, $y \in W$,
\[x + y \in W.\](Closure under scalar multiplication) For all $x \in W$ and $\lambda \in F$,
\[\lambda x \in W.\]
Let $V$ be a vector space and $\mathcal{W}$ be a collection of subspaces. Then $\bigcap \mathcal{W}$ is also a subspace.
Let $V$ be a vector space and $W$, $W’$ be subspaces. If $W \cup W’$ is a subspace, then either $W \subseteq W’$ or $W’ \subseteq W$.