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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?

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No, see the following paper of Gowers

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

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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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