Notation for Function Sequence Spaces

I wonder, is there any common notation about the function spaces.

For numerical sequences we have some notations such as

$c:= \{ (x_n) : (x_n)\,\ \text{convergent sequence} \}$

$c_0:= \{ (x_n) : (x_n) \,\,\text{convergent to}\,\, 0\}$

$l_\infty := \{ (x_n) : (x_n) \,\, \text{bounded} \}$

I need to use a common notation (if there is one I wanted to use it otherwise I am waiting for your suggestions.)

Let $X$ and $Y$ are nonempty sets and for all $n\in \mathbb{N}$, $f_n :X \to Y$ be functions.

(1) All function seqeunces that pointwisely (uniformly) bounded on the set $X$.

(2) All function sequence that pointwise convergent on the set X (at a point $x_0$).

(3) All function sequences that uniformly convergent on the set X.

Thanks in advance for your opinions and ideas.