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?

  • 1
    $\begingroup$ Try $f:c_0 \to \mathbb R$, $(x_n)_{n\in\mathbb N} \mapsto \sum\limits_{n=1}^\infty x_n/2^n$. $\endgroup$ – Jochen Feb 19 '15 at 11:47
  • $\begingroup$ @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? $\endgroup$ – Aubrey Feb 19 '15 at 14:10
  • $\begingroup$ You are right, I should have written this as an answer. $\endgroup$ – 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.