Let $T$ be a surjective, continuous linear operator between two Banach spaces $E$ and $F$. Assume that it is $B_F(y_0,4c)\subset \overline{T(B_E(0,1))}$, where $c>0$, $y_0 \in F$ ($B$ is for "ball", the line means "closure"). Using Minkowski addition, we get
$ B(0,4c) \subset \overline{T(B_E(0,1))} + \overline{T(B_E(0,1))}. $
Is $B(0,4c)$ included in the single $\overline{T(B_E(0,1))}$ too?