A sequence of functions fn converges to f uniformly on a set E if for any ε>0 there exists N such that for all n≥N and all x∈E, |fn(x) - f(x)| < ε. This is stronger than pointwise convergence, where N may depend on x. Uniform convergence implies the limit function is continuous if the fn are continuous, and allows termwise integration on a finite interval.
History
The rigorous definition appeared in 1870 in the works of Eduard Heine, based on Weierstrass's lectures. Weierstrass showed that the series ∑bk cos(akπx) for certain ak, bk converges uniformly, but its sum is nowhere differentiable — this shattered intuition about the connection between smoothness and convergence. Henri Poincaré in the 1880s used uniform convergence in the theory of automorphic functions and differential equations, and later the concept became part of the foundation of functional analysis through spaces C(X) with the uniform norm.
How it works
The key tool is estimation of the supremum. Weierstrass M-test: if |fn(x)| ≤ Mn and ∑Mn converges, then ∑fn converges uniformly (and absolutely). Uniform convergence is closely related to compactness: on a compact set, a pointwise convergent sequence may not be uniform, but from any uniformly bounded and equicontinuous sequence one can extract a uniformly convergent subsequence (Arzelà–Ascoli theorem). In numerical methods, uniform convergence ensures stability of approximation under parameter variation.
In practice
When designing an airplane wing, engineers expand flows into Fourier series; uniform convergence ensures that the approximation is equally good at all points of the profile.
Fun fact
The Fourier series of a continuous function can converge non-uniformly, and before Weierstrass's work, many believed this was impossible. The mistake cost several false proofs.
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