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Math Help - Complete subset

  1. #1
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    Complete subset

    Could someone give me a hand on this problem? I want to show that the set K=\{x=(x_n): \sum x_n=1\} in complete in C^n

    I tried to show that any convergent sequence in K must have its limit in C^n which means K is closed in a complete space, but I failed to do this.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Complete subset

    Quote Originally Posted by jackie View Post
    Could someone give me a hand on this problem? I want to show that the set K=\{x=(x_n): \sum x_n=1\} in complete in C^n
    Hint: K is compact and so sequentially compact.
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    Re: Complete subset

    Quote Originally Posted by FernandoRevilla View Post
    Hint: K is compact and so sequentially compact.
    Thanks a lot for your help, Fernando. However, I don't really see the connection between your hint and this problem yet. Also, I don't think I learn that sequentially compact implies complete. Or you want me show that K is compact in C^n, so it must closed.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Re: Complete subset

    Presumably C^n is \mathbb{C}^n. Note then that since \mathbb{C}^n is itself, you have that a given subspace is complete if and only if it's closed. But, evidently K is closed since the mapping (x_1,\cdots,x_n)\mapsto x_1+\cdots+x_n is a continuous (linear functional) map \mathbb{C}^n\to\mathbb{C}, and K is the preimage of the closed set \{1\}\susbseteq\mathbb{C} under this map.
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Re: Complete subset

    Quote Originally Posted by jackie View Post
    Also, I don't think I learn that sequentially compact implies complete.
    I immediately thought about a general result: Every compact metric space is complete. Drexel28's proposal is better.
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