Notation for the vector space of functions with $k$ continuous derivatives

Question

I saw the following definition given at the mathworld web site:

A function with $k$ continuous derivatives is called a $C^k$ function. In order to specify a $C^k$ function on a domain $X$, the notation $C^k(X)$ is used. The most common $C^k$ space is $C^0$, the space of continuous functions, whereas $C^1$ is the space of continuously differentiable functions.

My problem with the definition is that $C^k$ is presented as a function and also as a space. Can anybody help to explain this to me?

Answer 1

$C^k$ means space of functions on some domain, if written $C^k(X)$ then your domain is $X$, but if only $C^k$ that means domain is obvious from context. Some element $f \in$ $C^k$ means a $k$ continuous function. $C^k$ always means space