# How to interpret this notation

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

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.