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Definition 9.1. Let $f$ and $g$ be two maps. The $C^0$-distance between $f$ and $g$, written $d_0(f,g)$, is given by $$d_0(f,g)=\sup_{x\in\mathbb{R}}\vert f(x)-g(x)\vert$$ The $C^r$-distance $d_r(f,g)$ is given by $$d_r(f,g)=\sup_{x\in\mathbb{R}}\{\vert f(x)-g(x)\vert,\vert f'(x)-g'(x)\vert,\dots,\vert f^{(n)}(x)-g^{(n)}(x)\vert\}$$

How do I interpret this notation? $$\sup_{x\in\mathbb{R}}\sup_{1\ge i\ge n}\vert f^{(i)}(x)-g^{(i)}(x)\vert$$ or $$\sup_{1\ge i\ge n}\sup_{x\in\mathbb{R}}\vert f^{(i)}(x)-g^{(i)}(x)\vert$$ Are these equal? Also, when can you swap the $\sup$s? If both indexes run in a finite set then yes. What about other combinations? (finite-infinite or infinite-infinite)

This is from Robert L. Devaney's Introduction to Chaotic Dynamical Systems.

Thanks

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All it means is that you find the value of $x$ that maximizes the distance between two functions (or their derivatives, or second derivatives...).

You cannot always swap $sup$s, however, in the infinite case.

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