Problem
Suppose $U$ is a subspace of $V$. What is $U + U$?
Solution
By directly applying the definition of sums of subspaces: $$U + U \stackrel{D1.36}{=} \lbrace 2u | u \in U \rbrace$$ Now let $v = 2u$, then $v \in U$, because $U$ is a subspace and thus closed under multiplication. Therefore: $$U + U = \lbrace v | v \in U \rbrace = U$$