uniform convergence

一致收敛

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A sequence of real functions f_n on a common set E converges uniformly to f if for every epsilon > 0 some N makes |f_n(x)-f(x)| < epsilon for all n >= N and all x in E.

同一定义域上的实函数列若对每个正误差都有统一指标，使以后所有函数在每一点的误差均小于该值，则一致收敛。

Example

$f_n(x)=x/n$ on $[0,2]$ has supremum error $2/n\to0$.

Usage and distinctions

A pointwise convergence index may depend on x; a uniform index must work over the whole stated domain.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§3.1, Definition 3.1.9 · Checked 2026-10-14

Translation sources

1. 李东风（北京大学）— 金融时间序列分析备课笔记，附录EE.5 可测函数极限，一致收敛 · Checked 2026-10-14

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