11
$\begingroup$

Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. A Banach space $X$ over $K$ is stable if $X\cong X\times K$. I encountered the following question in some papers in the sixties:

Is every infinite dimensional Banach space stable?

Is this question still open?

$\endgroup$
0

2 Answers 2

13
$\begingroup$

No, see the following paper of Gowers

https://blms.oxfordjournals.org/content/26/6/523.full.pdf

$\endgroup$
12
$\begingroup$

This is the famous Banach's hyperplane problem that was solved in the negative by W. T. Gowers. There is a whole industry in Banach space theory concerning spaces which have even more peculiar properties (google for hereditrarily indecomposable Banach spaces).

$\endgroup$
1
  • 1
    $\begingroup$ Thank you for this answer. I can only accept one answer, and you have unfortunately lost the coin flip. $\endgroup$
    – Thomas Rot
    Mar 31, 2016 at 21:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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