A consequence of the open mapping theorem

Question

We let $f$ be a bounded and surjective linear map from the Banach space $X$ onto the Banach space $Y$ and put $$ r_0=\inf\{r: f(B^X(0,r)\supset B^Y(0,1)\}. $$ Using the open mapping theorem, I have shown that $0<r_0<\infty$ (in fact $||f||\geq 1/r_0$) and that $$ f(B^X(0,r_0)\supset B^Y(0,1). $$ But I'm still wondering if $$ f(\overline{B^X(0,r_0)}\supset \overline{B^Y(0,1)} $$ holds or not?

Answer 1

The example $f:c_0\to \mathbb R$, $(x_n)_{n\in\mathbb N} \mapsto \sum\limits_{n=1}^\infty x_n/2^n$ shows that the statement for the closed balls does not hold in general.