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Math Help - compactness of closed unit ball

  1. #1
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    compactness of closed unit ball

    Let l∞ be the space of bounded sequences of real numbers, endowed with the norm
    ∥x∥∞ = supn∈N |xn | , where x = (xn )n∈N .
    Prove that the closed unit ball of l∞ , B(0, 1) = {x ∈ l∞ ; ∥x∥∞ ≤ 1} , is not compact.

    I'm thinking about using the notion of sequential compactness, since every sequence Xn has an upper limit here, but I'm not sure if that would help much. Could anyone please give me a hint? Any input is appreciated!
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  2. #2
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    Quote Originally Posted by nngktr View Post
    Let l∞ be the space of bounded sequences of real numbers, endowed with the norm
    ∥x∥∞ = supn∈N |xn | , where x = (xn )n∈N .
    Prove that the closed unit ball of l∞ , B(0, 1) = {x ∈ l∞ ; ∥x∥∞ ≤ 1} , is not compact.

    I'm thinking about using the notion of sequential compactness, since every sequence Xn has an upper limit here, but I'm not sure if that would help much.
    Using sequential compactness is a good idea. For example, you could take x_n to have a 1 for the n'th coordinate and zeros everywhere else. Then show that the sequence (x_n) has no convergent subsequence in \ell^\infty (because \|x_n-x_m\|=1 whenever n\ne m).
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