How is $C_0^k(\mathbb R)$ defined?

Question

I'm sorry for asking such a short question, but I see the space $C_0^k(\mathbb R)$ everywhere being used without a rigorous definition.

$C_0(\mathbb R)$ is the space of continuous functions on $\mathbb R$ vanishing at infinity. The question is, is $C_0^k(\mathbb R):=C_0(\mathbb R)\cap C^k(\mathbb R)$ or is $C_0^k(\mathbb R):=\left\{f\in C^k(\mathbb R):f^{(i)}\in C_0(\mathbb R)\text{ for all }i\in\left\{0,\ldots,k\right\}\right\}$?

Those spaces shouldn't coincide.

Answer 1

Correct definition is that $C_0^k(\mathbb R):=\left\{f\in C^k(\mathbb R):f^{(i)}\in C_0(\mathbb R)\text{ for all }i\in\left\{0,\ldots,k\right\}\right\}$.

If you write $C_c(\Bbb R)$ set of all continuous function on $\Bbb R$ with compact support then $C_c^k(\mathbb R):=C_c(\mathbb R)\cap C^k(\mathbb R)$.