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Math Help - Compact Unit Ball Proof

  1. #1
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    Compact Unit Ball Proof

    Question:
    Let  \Vert \cdot      \Vert be any norm on  R^m and let  B = \{ x \in R^m  : ||x|| \le 1 \} . Prove that  B is compact. Hint: It suffices to show that  B is closed and bounded with respect to the Euclidean metric.

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    I don't think I'd have any difficulty doing this problem. I know that a subset of  R^m is compact if and only if it's closed and bounded, but I don't get why the hint says that it's sufficient to show that it's closed and bounded ONLY under the Euclidean metric. Isn't it possible that there may be other norms on  R^m such that the set is not closed or not bounded with respect to the other norm?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by JG89 View Post
    Question:
    Let  \Vert \cdot      \Vert be any norm on  R^m and let  B = \{ x \in R^m  : ||x|| \le 1 \} . Prove that  B is compact. Hint: It suffices to show that  B is closed and bounded with respect to the Euclidean metric.

    -------------------------------------------------------------------

    I don't think I'd have any difficulty doing this problem. I know that a subset of  R^m is compact if and only if it's closed and bounded, but I don't get why the hint says that it's sufficient to show that it's closed and bounded ONLY under the Euclidean metric. Isn't it possible that there may be other norms on  R^m such that the set is not closed or not bounded with respect to the other norm?
    No! That's the point of the problem : to show that any norm induces the same topology on Euclidean space.
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