Spectrum of Element of Banach Algebra is Compact
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Theorem
Let $\struct {A, \norm {\, \cdot \,} }$ be a Banach algebra over $\C$.
Let $x \in A$.
Let $\map {\sigma_A} x$ be the spectrum of $x$ in $A$.
Then $\map {\sigma_A} x$ is compact.
Proof
The result follows immediately from:
$\blacksquare$