4
$\begingroup$

Let $X$, $Y$, and $Z$ be Banach spaces with $Z \subset X$. Suppose $T$ is a bounded linear operator with domain $X$ and range $Y$. Must $T(Z)$ be a Banach space?

$\endgroup$

1 Answer 1

6
$\begingroup$

Take X to be $Z\oplus Y$, and suppose there is an injective map $S: Z \to Y$ which has non-closed range. Let $T(z,y) = S(z)-y$, then $T: X \to Y$ is surjective, but $T(Z) = S(Z)$ is not closed in $Y$.

$\endgroup$
1
  • 2
    $\begingroup$ To be a bit more explicit, let $Z = \ell^1$, $Y = \ell^2$ and let $S$ be the inclusion. $\endgroup$
    – t.b.
    Mar 15, 2012 at 4:18

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.