# What does this notation mean: $C([a,b], \mathbb{R})$, $C([a,b], \mathbb{R^m})$?

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?

• No, it's continuous functions whose domain is $[a, b]$ and codomain is $\mathbb{R}^m$. So each individual component is a function in $C([a, b], \mathbb{R})$. – user296602 Feb 22 at 19:12

$$C([a,b],\mathbb{R})$$ 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 $$C^0(X,Y)$$.