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I saw the post about a continuous mapping $F: X \to X$ where $X$ is an infinite-dimensional Banach space that maps the closed unit ball of $X$ onto the unbounded set.

A continuous mapping with the unbounded image of the unit ball in an infinite-dimensional Banach space

It is very interesting to me, is there such a continuous mapping $F: X \to X$ that maps the closed unit ball of $X$ onto its exterior.

That is, let, as in the previous link, $X$ be an infinite-dimensional Banach space, $B=\{x\in X: \|x\| \leq 1\}$, $E=\{x\in X: \|x\| \geq 1\}$.

Does there exist a continuous mapping $F: X \to X$ such that $F(B)=E$? Or, at least, $F(B)\subset E$.

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Yber597
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1 Answers1

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Theorem. If $X$ is an infinite dimensional Banach space then $B\subset X$ is homeomorphic to $E\subset X$.

Proof. First of all, $B$ is homeomorphic to $X$ (Theorem 6.2 in the book "Selected Topics in Infinite Dimensional Topology" by Bessaga and Pelczynski). On the other hand, the inversion $$ J: x\mapsto \frac{x}{|x|^2} $$ sends $E$ homeomorphically to $B-\{0\}$. Thus, it remains to show that $X$ is homeomorphic to $X-\{0\}$. This is a special case of Corollary 5.1 in the same book: For every compact subset $K\subset X$, the space $X$ is homeomorphic to $X-K$. qed

Moishe Kohan
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