Let $X$ be an algebra $C([0,1],\mathbb{R})$. Let define $F:X\ni f \mapsto f(0)f \in X$
What is the biggest $r$ such that $F$ is $C^r$-diffeomorphism ?
2026-03-28 15:55:30.1774713330
Diffeomorphism in Banach algebra
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$F$ sends all functions vanishing at $0$ to the zero functions, and is therefore in no way shape of form a diffeomorphism. What is the sup of the empty set ?... ;-)