# Math Help - Banach Algebra question

1. ## 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 $\|y-x\|<\delta$

2. Originally Posted by Mauritzvdworm
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 algebras—look in the index for "continuity of spectrum".