Let $E$ be a normed vector space. Let $U$ be open in $E$ and $V\subset E$. Prove that $U+V$ is open in $E.$
My trial:
Since $U$ is open in $E$, then $\forall x\in U$, $\exists \;r>0$ such that $B(x,r)\subset U.$
From here, I don't know how to proceed! Please, can someone help out with proof with balls??
Best Answer
Notice that $$U + V = \bigcup_{v \in V} (U + \{ v \} ).$$ Since each element in the union is a translate of the open set $U$, it is open. Thus, $U + V$ is open, as it is the union of open sets.