C[0,1] with the sup metric

1. Jul 18, 2008

coverband

in a question if you're asked to CONSIDER C[0,1] with the sup metric, does this mean that for the appearance of d(g,f) throughout this question the MAXIMUM distance between g and f (i.e. d_infinity) is to be considered?

Its like a way of saving writing d_infinity all the time if you just state at the start "consider with the sup metric" and have d alone throughout ?

2. Jul 18, 2008

CompuChip

They denote the metric (conventionally) by d and they mean that d is the sup metric (which, if more than one metric is relevant, is conventionally denoted $d_\infty$). So
$$d(f, g) = \sup_{x \in [0, 1]} |f(x) - g(x)|$$
(assuming the metric on [0, 1] is just the Euclidean one )

If that's what you meant, you're right.

3. Jul 18, 2008

coverband

So "consider C[0,1] with the sup metric..." means everytime you see "d" throughout the question, this means "d_infinity"

4. Jul 20, 2008

Pere Callahan

This means that whenever one refers to the metric of $C[0,1]$, be it by the symbol $d$ (provided this symbol is used in this context to denote the metric of $C[0,1]$) or otherwise, one means the sup metric, which is usually denoted $d_\infty$.

5. Jul 20, 2008

cool thank s