TITLE: Do non-stable Banach spaces exist?
QUESTION [11 upvotes]: 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?

REPLY [13 votes]: No, see the following paper of Gowers
https://blms.oxfordjournals.org/content/26/6/523.full.pdf

REPLY [12 votes]: 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).