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? 



Two vector spaces with homeomorphic open subsets are isomorphic?
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 infinite-dimensional Topology" for starters).


