compact linear operator

紧算子

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

Translation evidence: Literature usage

Meaning in context

A linear map between Banach spaces is compact if the image of every bounded set has compact closure in the codomain.

Banach 空间之间的线性映射若把每个有界集映为相对紧集，即其像的闭包紧，则为紧算子。

Example

On ℓ2\ell^2, Tx=(xn/(n+1))Tx=(x_n/(n+1)) is compact: truncation to n≤Nn\le N approximates it in operator norm with error 1/(N+2)1/(N+2).

Usage and distinctions

A bounded operator need not be compact; the identity on an infinite-dimensional Hilbert space is the basic counterexample.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§7.1, Definition 7.1.12 and Proposition 7.1.14 · Checked 2026-10-14

Translation sources

  1. 电子科技大学 — 泛函分析教学大纲第2章，2.4 紧算子；了解1 · Checked 2026-10-14
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