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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Existence of a mapping in a nonseparable Banach space that moves all nearby points to far-away points

Does there exist a nonseparable Banach space $X$, a mapping $F: X\to X$, and an open nonempty subset $D\subset X$ such that $$ \forall\,E>0 \quad \exists\,\delta>0: \quad \forall\,x,y\in D \quad ...