A Nontopological Proof of the Uniform Boundedness Theorem

Abstract

The purpose of this article is to present a proof of the uniform boundedness theorem which can be comprehended by observing a single equation. The proof is elementary in Halmos's sense of the word in that it does not use the Baire Category theorem or any related lemmas. In fact, this proof was prompted by Halmos's [1, Problem 20], which asks for an elementary proof in the special case of Hilbert space.