# A consequence of the open mapping theorem

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?

• Try $f:c_0 \to \mathbb R$, $(x_n)_{n\in\mathbb N} \mapsto \sum\limits_{n=1}^\infty x_n/2^n$. – Jochen Feb 19 '15 at 11:47
• @Jochen Great example, thanks so much!! I'm not sure if I should write it up as an answer below or delete this question or simply leave it as it is? – Aubrey Feb 19 '15 at 14:10
• You are right, I should have written this as an answer. – Jochen Feb 19 '15 at 14:31

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.