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Metric Spaces

  1. Mar 5, 2009 #1
    Let X be a set, and let fn : X [tex]\rightarrow[/tex] R be a sequence of functions. Let p be the uniform metric on the space Rx. Show that the sequence (fn) converges uniformly to the function f : X [tex]\rightarrow[/tex] R if and only if the sequence (fn) converges to f as elements of the metric space
    (RX, p)

    I'm not sure how to do this problem...

    Can someone help me out?

    Thanks!
     
  2. jcsd
  3. Mar 5, 2009 #2
    Where are you stuck? Start by writing down all the relevant definitions (uniform metric, uniform convergence, convergence in metric spaces) and you will be almost done.
     
  4. Mar 7, 2009 #3
    please define what is mean't by " uniform metric"
     
  5. Mar 7, 2009 #4
    It's the metric obtained from the http://en.wikipedia.org/wiki/Uniform_norm" [Broken], so

    [tex]d(f,g)=\sup_{x\in X}|f(x)-g(x)|[/tex]

    where [tex]f,g:X\to\mathbb{R}[/tex].
     
    Last edited by a moderator: May 4, 2017
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