The main existence theorems in calculus are the Intermediate Value Theorem, the Extreme Value Theorem, Rolle's Theorem, and the Mean Value Theorem.

He slighted the plaudits of their theatre, he abhorred their dances and their horse-races, he was abstinent even at a festival, and incorrupt himself, perpetually admonished the dissipated citizens of their impious abandonment of the laws of their country.

Paragraph five deals with (A*)-algebras and contains a proof of a vector-valued variant of Wiener's Tauberian theorem.

If you can't do this then there's fuck all hope for you.