Uniform boundedness principle

The uniform boundedness principle (or Banach-Steinhaus theorem) states that a family of continuous linear operators between Banach spaces which is pointwise bounded — bounded at every point of the domain — is automatically uniformly bounded in operator norm. This prevents the pathological scenario where a sequence of operators converges pointwise but whose norms diverge to infinity, a phenomenon that would destroy the interchange of limits in analysis. The principle is the foundation of convergence theorems for Fourier series, the stability theory of numerical methods, and the theory of operator semigroups. Like the open mapping theorem, its proof relies on the Baire category theorem, and this shared dependence reveals that the three great theorems of functional analysis are not separate results but manifestations of a single topological fact: complete normed spaces are not meager in themselves.