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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.

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    @Beat: Assuming that that comment was intended to refer to my answer: Yes, that's how I understood your question, and that's what I answered. This is false for the reasons I $p$ointed out.2011-12-12

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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.