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A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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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 …
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