pointwise convergence

点点收敛

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 E converges pointwise to f if for each fixed x in E the scalar sequence f_n(x) converges to f(x).

实函数列在集合 E 上点点收敛，是指固定任一点 x 后，数列 f_n(x) 收敛于 f(x)；所需指标可以随 x 改变。

Example

xnx^n on [0,1)[0,1) tends pointwise to zero, although its supremum is always 11.

Usage and distinctions

Pointwise convergence need not preserve continuity or imply uniform convergence.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§3.1, Definition 3.1.10 · 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，点点收敛 · Checked 2026-10-14
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