open mapping theorem for Banach spaces

开映射定理

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Translation evidence: Literature usage

Meaning in context

A bounded surjective linear map between Banach spaces sends every open set in its domain to an open set in its codomain.

Banach 空间之间的有界线性满射，把定义域中的每个开集映成值域空间的开集。

Example

T:R2→RT:\mathbb R^2\to\mathbb R, T(x,y)=xT(x,y)=x, maps the open unit disk to (−1,1)(-1,1).

Usage and distinctions

Surjectivity is essential: inclusion of a line into R^2 is continuous but its image of an open interval is not open in R^2.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§6.3, Theorem 6.3.4 · 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2及重点与难点：开映射定理 · Checked 2026-10-14
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