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?


No, see the following paper of Gowers



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).

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    $\begingroup$ Thank you for this answer. I can only accept one answer, and you have unfortunately lost the coin flip. $\endgroup$ – Thomas Rot Mar 31 '16 at 21:24

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