weak convergence in a normed space

弱收敛

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

Translation evidence: Literature usage

Meaning in context

A sequence x_n in a normed space converges weakly to x if lambda(x_n) tends to lambda(x) for every bounded linear scalar functional lambda.

赋范空间中的序列若经每个连续线性泛函测量后的标量列都趋于该泛函在 x 处的值，则弱收敛于 x。

Example

In $\ell^2$, the unit vectors $e_n$ converge weakly to $0$ since every square-summable sequence has nth coordinate tending to zero.

Usage and distinctions

Norm convergence implies weak convergence; the converse fails because $\|e_n\|_2=1$.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§6.7, Definition 6.7.11 · 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.10.2 性质，弱收敛 · Checked 2026-10-14

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