closed graph theorem

闭图像定理

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

Translation evidence: Literature usage

Meaning in context

An everywhere-defined linear map T between Banach spaces is bounded if its graph {(x,Tx)} is closed in their product norm topology.

在两个 Banach 空间之间处处有定义的线性映射，若其图像在乘积范数拓扑中为闭集，则该映射有界。

Example

The coordinate swap T(x,y)=(y,x)T(x,y)=(y,x) has a closed graph in R2×R2\mathbb R^2\times\mathbb R^2 and norm 11.

Usage and distinctions

An operator defined only on a proper dense domain may be closed and unbounded; the theorem requires the full Banach domain.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§6.4, Theorem 6.4.2 · 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：闭图像定理 · Checked 2026-10-14
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