# Banach conjecture

There was a paper today in the arXiv mailing list (arXiv:2006.00336) proving yet another case of the Banach conjecture. I never heard of this conjecture before, but it is easy to state and seems to me to be a foundational recognition principle for those Banach spaces that are actually Hilbert spaces.

The conjecture was stated by Stefan Banach in 1932 and reads as follows: if $$V$$ is a Banach space (it may be real or complex, and of finite or infinite dimension) such that for some natural number $$n$$ with $$2 \le n < \mathrm{dim}(V)$$ all $$n$$-dimensional subspaces of $$V$$ are pairwise isometrically isomorphic to each other, then $$V$$ is a Hilbert space.

Many cases of the Banach conjecture are already known (for example, the $$\infty$$-dimensional case in both the real and the complex version). The above cited arXiv preprint summarizes all the single results in its introduction – so have a look if you are interested.