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I know $C$ is for continuous functions, and I assume $([a,b], \mathbb{R})$ means an interval on the real number line, and for $([a,b], \mathbb{R}^m)$ I assume this means that you have some vector in $\mathbb{R}^m$ whose components can assume any value in the interval $[a,b]$ on the real number line. Correct?

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    $\begingroup$ No, it's continuous functions whose domain is $[a, b]$ and codomain is $\mathbb{R}^m$. So each individual component is a function in $C([a, b], \mathbb{R})$. $\endgroup$ – user296602 Feb 22 at 19:12
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$C([a,b],\mathbb{R})$ means continuous functions $f:[a,b]\to \mathbb{R}$. More generally, $C([a,b],\mathbb{R}^m)$ denotes continuous functions $f:[a,b]\to \mathbb{R}^m$. Even more generally, given two topological spaces $X$ and $Y$, $C(X,Y)$ denotes continuous functions from $X$ to $Y$. You might also see the notation $C^0(X,Y)$.

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