bounded linear operator

有界线性算子

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

Translation evidence: Literature usage

Meaning in context

A linear map T between normed spaces is bounded when some finite C satisfies norm(Tx)<=C norm(x) for every domain vector x.

赋范空间之间的线性映射若有一个有限常数 C，使每个向量的像范数不超过 C 倍原范数，则为有界线性算子。

Example

For $T(x,y)=(2x,-y)$ in Euclidean norm, $\|T(x,y)\|\le2\|(x,y)\|$.

Usage and distinctions

A bounded linear operator maps bounded sets to bounded sets, but its image on the entire nonzero vector space is usually unbounded.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§3.2, Definition 3.2.1 and Theorem 3.2.5(f) · Checked 2026-10-14

Translation sources

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

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