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!

How to prove the following defined metric space is separable

  1. Oct 1, 2015 #1
    • Member warned about posting without the homework template
    Let ##\mathbb{X}## be the set of all sequences in ##\mathbb{R}## that converge to ##0##. For any sequences ##\{x_n\},\{y_n\}\in\mathbb{X}##, define the metric ##d(\{x_n\},\{y_n\})=\sup_{n}{|x_n−y_n|}##. Show the metric space ##(\mathbb{X},d)## is separable. I understand that I perhaps need to find a countable dense set in ##\mathbb{X}##.
     
  2. jcsd
  3. Oct 1, 2015 #2

    Krylov

    User Avatar
    Science Advisor
    Education Advisor

    Depending on whether or not this is an exercise question, you may be able to give a quick answer by noting that the dual space of ##X## is ##\ell_1##.
     
  4. Oct 1, 2015 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, that's exactly what you should do. Any thoughts? To get started do you know a countable dense subset of ##\mathbb{R}##?
     
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: How to prove the following defined metric space is separable
  1. Prove metric space (Replies: 2)

Loading...