# What is Banach's Homogeneous Space Problem of 1932?

I have seen some references to Banach's homogeneous space problem from 1932 which was solved by Gowers.

Could I ask what this problem actually states, I have tried searching and cannot find anything (or nothing which was free access anyway).

For me, the very first Google hit for "Banach homogeneous space problem" is https://arxiv.org/abs/math/9205207, which is open access. It clearly states the question and describes the progress up to 1992:

A Banach space is said to be homogeneous if it is isomorphic to all of its infinite dimensional subspaces. Is every homogeneous Banach space isomorphic to a Hilbert space?

(In context, here "$$X$$ and $$Y$$ are isomorphic" means "there is a bijective linear homeomorphism between them".)

Gowers' proof appears to be in the following paper:

Gowers, W. T. An infinite Ramsey theorem and some Banach-space dichotomies. Ann. of Math. (2) 156 (2002), no. 3, 797–833. https://doi.org/10.2307/3597282

It is also freely available on arXiv at https://arxiv.org/abs/math/0501105.

• That's not my first hit when I google 'Banach homogeneous space problem', maybe your google works differently based on what you usually search for. – Tom Sep 23 '19 at 10:15