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?
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$\begingroup$ 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. $\endgroup$ – Simon Rose Sep 30 '15 at 9:49

$\begingroup$ My guess is that it is true in all spaces, since it is true for finitedimentional ones. But I cannot give a proof or a disproof. $\endgroup$ – usr203050 Sep 30 '15 at 9:51

1$\begingroup$ 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. $\endgroup$ – usr203050 Sep 30 '15 at 10:06

1$\begingroup$ 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. $\endgroup$ – shasta Sep 30 '15 at 10:19
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This is false. All separable Banach spaces, for example, are homeomorphic.Indeed, there is a considerable body of work on when topological vector spaces are homeomorphic (see Bessaga and Pelczynski "Selected Topics in infinitedimensional Topology" for starters).

$\begingroup$ Ok thank you. So it is far from evident in any case. $\endgroup$ – usr203050 Sep 30 '15 at 9:59

$\begingroup$ And what if the subsets are smoothly diffeomorphic? (The second question). $\endgroup$ – usr203050 Sep 30 '15 at 10:03

1$\begingroup$ Isn't the diffeomorphic part answered by the comment of usr203050? $\endgroup$ – Jochen Wengenroth Sep 30 '15 at 10:17

2$\begingroup$ If diffeomorphism is something that has differential as a bounded invertible operator between tangent spaces, then this very operator is linear isomorphism between spaces. If not, what concretely do you mean? $\endgroup$ – Fedor Petrov Sep 30 '15 at 10:17

$\begingroup$ That is what I mean! :) Also, the comment of shasta gives something more. $\endgroup$ – usr203050 Sep 30 '15 at 10:29