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C[0,1] with the sup metric

  1. Jul 18, 2008 #1
    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. jcsd
  3. Jul 18, 2008 #2

    CompuChip

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    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 [itex]d_\infty[/itex]). So
    [tex]d(f, g) = \sup_{x \in [0, 1]} |f(x) - g(x)|[/tex]
    (assuming the metric on [0, 1] is just the Euclidean one :smile:)

    If that's what you meant, you're right.
     
  4. Jul 18, 2008 #3
    So "consider C[0,1] with the sup metric..." means everytime you see "d" throughout the question, this means "d_infinity"
     
  5. Jul 20, 2008 #4
    This means that whenever one refers to the metric of [itex]C[0,1][/itex], be it by the symbol [itex]d[/itex] (provided this symbol is used in this context to denote the metric of [itex]C[0,1][/itex]) or otherwise, one means the sup metric, which is usually denoted [itex]d_\infty[/itex].
     
  6. Jul 20, 2008 #5
    cool thank s
     
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