norm on a vector space

范数

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

Translation evidence: Literature usage

Meaning in context

A norm on a real or complex vector space is a nonnegative real function N with N(x)=0 exactly for x=0, N(alpha x)=|alpha|N(x), and N(x+y)<=N(x)+N(y).

实或复线性空间上的非负函数，若仅在零向量取零、满足绝对齐次性及三角不等式，则为范数。

Example

$\|(x,y)\|_1=|x|+|y|$ is a norm and $\|(3,-2)\|_1=5$.

Usage and distinctions

A seminorm may vanish on nonzero vectors; squared Euclidean length is not a norm because it scales quadratically.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§1.2, Definition 1.2.13 · 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.9.2 距离空间，范数三条公理 · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.