The Principle of Uniform Boundedness

Abstract

Many of the most important theorems in analysis assert that pointwise hypotheses imply uniform conclusions. Perhaps the simplest example is the theorem that a continuous function on a compact set is uniformly continuous. The main theorem in this section concerns a family of bounded linear operators, and asserts that the family is uniformly bounded (and hence equicontinuous) if it is pointwise bounded. We begin by defining these terms precisely.