My question is this, is there a proof to show that suppose the distance $||h(x)- g(x)||< 4$, then $|h(x) - g(x) | <4$ for all $x\in [-\pi, \pi]$? I know from Schwarz inequality that $$|h - g| \leq ||h- g||.$$
Note that the inner product space $PC [-\pi , \pi]$ the distance between two functions is $$ ||h-g||^2= \int_{-\pi}^{\pi} |h(x) - g(x) |^{2} dx.$$ I hope I can just state that and there will not be anything to prove.
No, this is not the case. Given your definition of the distance between two functions, the function values can be arbitrarily far apart if this occurs in a sufficiently small region. For instance, consider $g(x)=0$ and $h(x)$ a rectangular pulse that can be arbitrarily high as long as it is sufficiently short.