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Show subspace of normed vector is closed under sup norm.

  1. Oct 15, 2012 #1
    http://imageshack.us/a/img141/4963/92113198.jpg [Broken]

    hey,

    I'm having some trouble with this question,

    For part a) I know that in order for c_0 to be closed every sequence in c_0 must converge to a limit in c_0 but I am having trouble actually showing that formally with the use of the norm given.

    Say x_n is a sequence in c_0 which converges to x then for any ε>0 there exists an N such that whenever n>N |x_n - x| < ε.

    So that's just the definition for the limit of a sequence, but I'm having trouble how to use this to show that the limit x is in c_0 for every x_n in c_0.

    Would anyone be able to help me out?

    Thanks in advance
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Oct 15, 2012 #2

    micromass

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    So, you wish to show that [itex]x=(x^0,x^1,x^2,x^3,...)\in c_0[/itex]. Take [itex]\varepsilon>0[/itex]. You'll need to find an N>0 such that for all n>N holds that [itex]|x^n|<\varepsilon[/itex].

    So, what you need to do is make [itex]|x^n|[/itex] smaller than something that is smaller than [itex]\varepsilon[/itex]. What could you make [itex]|x^n|[/itex] smaller as? Hint: use the triangle inequality to introduce the [itex]x_k[/itex].
     
  4. Oct 15, 2012 #3
    Oh so using the trick where you add 0,

    |x_n|=|x_n - x_k + x_k|≤ |x_n - x_k|+|x_k|

    Then for x_k in c_0,

    For ε>0 there is an N > 0 such that for all n>N, |x_k| <ε/4

    So

    |x_n - x_k|+|x_k| < |x_n - x_k|+ε/4

    Then |x_n - x_k| will definitely be smaller then ε so

    |x_n - x_k|+ε/4 < ε


    I am having trouble seeing where the sup norm comes in,

    Have I missed something?
     
  5. Oct 17, 2012 #4
    Does that look ok?

    That shows that x_n is Cauchy, since you have

    ||x_n - x_k|| < 3e/4 < e

    for n,k > N = max(N_n, N_K)
     
    Last edited: Oct 17, 2012
  6. Oct 17, 2012 #5
    Do you mean by [tex]x= \{ x_1,x_2,..\} [/tex] that x_1 is a sequence of itself?

    So for example [tex]x_1 (n) = 0.5^n, n =1, 2, 3,...[/tex]
    [tex]x_2(n) = 0.4^n, n =1, 2, 3,...[/tex]

    Or is x just ONE sequence?
     
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