Let $V$ be a [[vector space]] over the [[field]] $K$. Let $U, W$ be [[subspace|subspaces]] of $V$. The *sum* of $U$ and $W$ is the [[subset]] $(U+W)$ of $V$ containing the sums $u+w$ with $u \in U$ and $w \in W$. $(U+W)=\set{u+w:(u\in U\ \land\ w\in W)}$ It is also a subspace of $V$. $u_{1}, u_{2} \in U$ and $w_{1}, w_{2} \in W$ then - $(U+W)$ is closed under addition, - $\left(u_{1}+w_{1}\right)+\left(u_{2}+w_{2}\right)=u_{1}+u_{2}+w_{1}+w_{2} \in (U+W)$ - $(U+W)$ is closed under scalar multiplication - $c \in K \implies c\left(u_{1}+w_{1}\right)=c u_{1}+c w_{1} \in (U+W)$ - $(U+W)$ contains 0 - $U,W\ni 0$, therefore $0+0=0\in (U+W)$ We only call $(U\oplus W)$ a **direct sum** in the special case that there is a unique $u,w$ pair that sum to to the element $(u+w)$