I am taking functional analysis class and we stated the Arzela-Ascoli theorem in the following way:
Let $(K, d)$ be a compact metric space, $(Y, \| · \|_{Y} )$ a Banach space and $\mathcal{F} \subseteq C(K; Y)$ be a subset of functions in $C(K; Y)$. Then $\mathcal{F}$ is precompact if and only if the following two conditions hold.
(i) $\mathcal{F}$ is equicontinuous.
(ii) $\mathcal{F}$ is pointwise precompact i.e. the set $\mathcal{F}_{x} := \{f(x) | f ∈ \mathcal{F}\} \subseteq Y$ is precompact in $Y$ $∀x ∈ K$.
However, when I looked online I found other versions of the theorem. In particular, I have seen somewhere (I am not sure where) that instead of (ii) said $\mathcal{F}$ should be pointwise bounded. Is pointwise boundedness (or uniform boundedness for that matter) equivalent to (ii)?