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Math Help - Banach algebras question 6

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    Banach algebras question 6

    Here is the last my question in this theme.

    I've problems with the following statements in red shapes. How can I show/explain this in the easiest way?
    Where we used fact that this measure is Radon measure?




    PS. Here is link to the original paper http://matwbn.icm.edu.pl/ksiazki/sm/sm29/sm29126.pdf
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    Quote Originally Posted by Arczi1984 View Post
    Here is the last my question in this theme.

    I've problems with the following statements in red shapes. How can I show/explain this in the easiest way?
    Where we used fact that this measure is Radon measure?




    PS. Here is link to the original paper http://matwbn.icm.edu.pl/ksiazki/sm/sm29/sm29126.pdf
    A Radon measure is just a measure for which the formula on the left side of (6) defines a continuous linear functional.

    In a function algebra, the spectrum of a function is just the range of the function. So the formula (6) ensures that f(x) is in the spectrum of x.

    In a compact space, every sequence has a convergent subsequence. So some subsequence of the sequence (p_n) converges to a limit p, and (since x is continuous) the condition |x(p_n)-f(x)|\to0 implies that x(p) = f(x).
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