= Nonemptiness of the Banach-algebra spectrum
Every element of a nonzero complex unital <Banach algebra> has a nonempty compact <spectrum of an element>. If the <resolvent of an element> existed everywhere, composing it with any <bounded linear functional> would give a bounded <entire function> tending to zero at infinity. The <Liouville theorem> and point separation by the <Hahn-Banach theorem> would force the resolvent to be zero, contradicting the inverse equation. Nonunital algebras are treated by <unitization>.
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