What is the meaning of this symbol ? $a)$ $C_b^0(\mathbb{R})$ $b)$ $C_c^{\infty}(\mathbb{R})$.

Question

What is the meaning of this symbol ?

$a)$ $C_b^0(\mathbb{R})$

$b)$ $C_c^{\infty}(\mathbb{R})$.

actually im asking this question because i found the answer Which of the following subsets are dense in the given spaces?

i didn't understand the option $b)$

Notes: i have seen the user those who have been answer have not active since for 20 days ,, so i decide to upload a new post

thanks u

Answer 1

According to the question,

1. space of bounded [the foot $b$] continuous functions [$\mathcal C^0$] defined on $\Bbb R$.

2. space of infinitely differentiable functions [$\mathcal C^\infty$] with compact support [the foot $c$] defined on $\mathbb R$.

support: the closure of the set $\{x \in \mathbb R\colon f(x) \neq 0\}$.