Hahn–Banach extension theorem

Hahn-Banach定理

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

Translation evidence: Literature usage

Meaning in context

A bounded linear scalar functional on any linear subspace of a real or complex normed space extends to the whole space with the same norm.

实或复赋范空间任意线性子空间上的有界线性泛函，都可延拓到整个空间且保持范数不变。

Example

On the x-axis of Euclidean R2\mathbb R^2, λ(x,0)=x\lambda(x,0)=x extends as Λ(x,y)=x\Lambda(x,y)=x, both of norm 11.

Usage and distinctions

Completeness and closedness of the subspace are unnecessary; this scalar-functional theorem does not assert the same extension for arbitrary vector-valued maps.

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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§3.8, Theorem 3.8.5 · Checked 2026-10-14

Translation sources

  1. 电子科技大学 — 泛函分析教学大纲第2章，2.2 对偶与Hahn-Banach定理；掌握4 · Checked 2026-10-14
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