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.