1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Showing a metric space is complete

  1. May 5, 2014 #1
    1. The problem statement, all variables and given/known data
    Show the space of all space of all continuous real-valued functions on the interval [0, a] with the metric [itex]d(x,y)=sup_{0\leq t\leq a}e^{-Lt}|x(t)-y(t)|[/itex] is a complete metric space.

    3. The attempt at a solution
    Spent a few hours just thinking about this question, trying to prove it directly from the definition that says a complete metric space is one where every Cauchy sequence in it has a limit in the space.

    I started with an arbitrary Cauchy sequence of functions [itex]d(x_m, x_n)=sup_{0\leq t\leq a}e^{-Lt}|x_m(t)-x_n(t)|[/itex]...that's it! I don't know how to find this limit and show that the sequence converges to that.
     
  2. jcsd
  3. May 5, 2014 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Try to show that your sequence converges pointswise. So show that for every ##t##, the sequence ##(x_n(t))_n## converges to some real number which I denote by ##x(t)##. Then show that ##x## defined like this is continuous and that ##x_n\rightarrow x## in your metric.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Showing a metric space is complete
  1. Complete metric space (Replies: 5)

  2. COMPLETE metric space (Replies: 28)

Loading...