# Two vector spaces with homeomorphic open subsets are isomorphic?

Is it true that if $E,F$ are two topological vector spaces (or say Banach spaces) over $\mathbb{R}$ such that they have nonempty open subsets $U\subset E, V\subset F$ which are homeomorphic, then the two vector spaces are isomorphic? If false, then what can we say if the two open subsets are $\mathcal{C}^1$-diffeomorphic?

• I would guess that the first question is false, simply on the grounds that it is pretty trivially true for finite dimensional ones, so it's probably false for arbitrary vector spaces. – Simon Rose Sep 30 '15 at 9:49
• My guess is that it is true in all spaces, since it is true for finite-dimentional ones. But I cannot give a proof or a disproof. – usr203050 Sep 30 '15 at 9:51
• My guess for the second part is that the differential at some point (or may be at any point) can give an isomorphism between the two spaces. – usr203050 Sep 30 '15 at 10:06
• Yes. The interesting question between those two extremes is the case of Lipschitz equivalence (for Banach spaces). Again much work has been done on this case, e.g. by Lindenstrauss et al. – shasta Sep 30 '15 at 10:19