Uniform convergence

A sequence of functions converges uniformly when one index threshold makes the approximation error arbitrarily small at every point of the domain. The threshold is independent of the input.

Supremum norm: The supremum norm of a function is the supremum of its absolute value over a specified domain. Uniform error is measured by the supremum norm of the difference between each function and its limit.
Pointwise convergence: A sequence of functions converges pointwise when it converges at each fixed domain point, possibly with different thresholds. Unlike uniform convergence, its threshold may vary with the input.
Term-by-term integration: Term-by-term integration integrates a function series by integrating its individual terms. Uniform convergence on a suitable interval provides conditions for integrating the limit through the sequence.
Sequence of functions: A sequence of functions is an ordered list of functions sharing a specified domain and codomain. Uniform convergence describes how such a sequence approaches a single function.
Epsilon-N definition: A convergence definition requiring that, for every positive error tolerance, all terms past one index satisfy the bound. Its quantifier order makes the shared threshold across all domain points explicit.
Convergence in measure: A sequence of measurable functions converges in measure when the measure of points with large error tends to zero. It controls the size of the exceptional set rather than requiring small error everywhere.
Term-by-term differentiation: Term-by-term differentiation differentiates a function series by differentiating its individual terms. Uniform convergence of derivative series, with suitable additional conditions, can justify this exchange.
Metric space: A metric space is a set equipped with a distance function satisfying the metric axioms. Uniform convergence can be expressed as convergence under the supremum metric when that metric is finite.
Uniform Cauchy criterion: A criterion stating that a function sequence converges uniformly when its terms become uniformly Cauchy. It tests uniform convergence using pairwise differences, without first specifying a limit function.
Almost-everywhere convergence: A sequence converges almost everywhere when it converges pointwise outside a set of measure zero. It permits a negligible exceptional set, whereas uniform convergence bounds error across the full domain.

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