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Math Help - Banach Algebra question

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    Member Mauritzvdworm's Avatar
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    Banach Algebra question

    Show that for every open set U\subset\mathbb{C} that contains the spectrum \sigma(x) of x\in A where A is a Banach algebra that there exists a \delta>0 such that \sigma(y)\subset U whenever y\in A satisfies \|y-x\|<\delta
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    Quote Originally Posted by Mauritzvdworm View Post
    Show that for every open set U\subset\mathbb{C} that contains the spectrum \sigma(x) of x\in A where A is a Banach algebra that there exists a \delta>0 such that \sigma(y)\subset U whenever y\in A satisfies \|y-x\|<\delta
    This is Theorem 1.6.16 in Rickart's Banach algebras. It should be in any respectable book on Banach algebrasólook in the index for "continuity of spectrum".
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