Consider the following problem. If I have Banach spaces $X,Y$, and an operator $T\in B(X,Y)$, how would I show that,
"If $T\in B(X,Y)$, and $p\in\overline{ T(B_{X,1}(0))}$ then $-p\in\overline{ T(B_{X,1}(0))}$."
I understand that I am trying to show that $\overline{ T(B_{X,1}(0))}\subset Y$ is convex, for $-p,p\in \overline{ T(B_{X,1}(0))}$, and I also understand that $\overline{ T(B_{X,1}(0))}$ is a closed set in $Y$ (since we are dealing with the closure).
However, I am not too sure where the point about $T$ being a bounded linear operator comes in. Could somebody please give me some direction for this problem?