Definition

If $U_1,U_2\subset V$ where V is a vector space, then we define the sum $U_1+U_2\subset V$ to be $$\begin{align*} U_1+U_2:=\{u_1+u_2|u_1\in U_1 \text{ and } u_2\in U_2\} \end{align*}$$ More generally, for $U_1,....U_m\subset V$ then $$\begin{align*} U_1+...+U_m:\{u_1+...+u_m|u_i\in U_i, i=1,...,m\} \end{align*}$$