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 $W’\subseteq W$.