In my textbook, the Arzela-Ascoli Theorem states that for a compact set $A$ in a metric space $M$ and $B\subset\mathcal{C}\left(A, N\right)$, where $N$ is another metric space, then $B$ is compact if and only if $B$ is closed, equicontinuous, and pointwise compact.
I also see two corollaries in my book that state "for a compact set $A\subset M$, $N=\mathbb{R^{n}}$, and an equicontinuous and pointwise bounded set $B\subset\mathcal{C}\left(A, \mathbb{R^{n}}\right)$, then every sequence in $B$ has a uniformly convergent subsequence" and "If $A\subset M$ is compact and $N=\mathbb{R^{n}}$ then $B\subset\mathcal{C}\left(A, \mathbb{R^{n}}\right)$ is compact if and only if it is bounded, closed, and equicontinuous."
I have a couple of questions:
$\left(1\right)$ What is pointwise compactness?
$\left(2\right)$ What is the difference between boundedness, pointwise boundedness, and uniform boundedness?
$\left(3\right)$ If my function is bounded for all $x$, does this imply that it is pointwise bounded at all $x$?
$\left(4\right)$ Does pointwise convergence imply pointwise boundedness?
For $\left(3\right)$, I think that if it the set $B$ is bounded, then for all $x$ it is bounded, so for each fixed $x$, it is pointwise bounded. But I am not sure if $x$ should really be $f$ in this case or not since it is a set of functions.
For $\left(4\right)$, I want to say that this is true because if it is pointwise convergent, then it must be bounded at that point.
The rest I am unsure of.