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Math Help - Banach algebras question 1

  1. #1
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    Banach algebras question 1

    I've questions about two steps in the proof of the following theorem:


    Red shape: From where we know that this functional is continuous?
    Blue shape: From where we know that completion of A is a Sub-norm algebra?

    PS. Here is link to the original paper: http://matwbn.icm.edu.pl/ksiazki/sm/sm29/sm29126.pdf
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  2. #2
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    Red: If f(x) = x(p_x) then |f(x)|\leqslant \|f\|_{\text{sup}}. Thus the linear functional f is bounded and hence continuous for the sup norm.

    Blue: The completion of any normed algebra is a normed algebra for that norm.
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