operator norm

线性算子范数

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

Translation evidence: Literature usage

Meaning in context

For a bounded linear map T between normed spaces, its operator norm is sup{norm(Tx):norm(x)<=1}, equivalently the least constant in its global linear bound.

有界线性映射的算子范数，是单位闭球中所有向量之像范数的上确界，也就是线性界中最小的常数。

Example

For $T(x,y)=(2x,-y)$ on Euclidean $\mathbb R^2$, $\|T\|=2$, attained at $(1,0)$.

Usage and distinctions

The matrix entry maximum is generally different; the operator norm depends on the norms chosen for domain and codomain.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§3.2, Proposition 3.2.6 and Definition 3.2.7 · Checked 2026-10-14

Translation sources

1. 电子科技大学 — 泛函分析教学大纲第2章，掌握1：线性算子范数 · Checked 2026-10-14

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