Banach space

Banach空间

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

Translation evidence: Literature usage

Meaning in context

A Banach space is a real or complex normed vector space in which every Cauchy sequence for the norm distance converges to an element of the space.

Banach 空间是对范数所诱导的距离完备的实或复赋范线性空间，即每个基本列都有属于该空间的范数极限。

Example

C([0,2])C([0,2]) with ∥f∥∞=max⁡∣f∣\|f\|_\infty=\max|f| is Banach: a uniformly Cauchy sequence has a continuous uniform limit.

Usage and distinctions

Completeness depends on the norm; the same continuous-function space with its integral norm is incomplete.

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Definition sources

  1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§4.1, Definition 4.1.1 and Example 4.1.5 · Checked 2026-10-14

Translation sources

  1. 电子科技大学 — 泛函分析教学大纲第一章赋范线性空间，1.2 Banach空间 · Checked 2026-10-14
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