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Co is closed in l∞

  1. Apr 2, 2009 #1
    I want to show the Banach space co is closed in l∞ .

    So, I pick a convergent sequence x_n in co that converges to x in l∞
    Now, x_n --> x: given e>0, there is an N_e s.t. for all n>N_e,
    ||x_n -x ||= Sup |x_n(k)-x(k)|<e (we're supping over k).

    Since x_n is a sequence in co , for each fixed n, x_n(k)-->0 as k--> infinity.
    So, given e>0, there is a K depending on n and e, such that for all k> K, we have |x_n(k)|<e.

    We want to show x is in co
    so we show there is a Ko such that for all k>Ko, |x(k)|<e

    I am having trouble getting this Ko.

    I know |x(k)|≤ |x_n(k)| + |x_n(k)-x(k)|≤ |x_n(k)| + Sup (over k) |x_n(k)-x(k)|

    we have |x_n(k)|<e as k>K, but Sup (over k) |x_n(k)-x(k)|<e for n>N_e.


    So I am not so sure how to get this Ko.
     
  2. jcsd
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