##  [Open Mapping Theorem](/index.php/open-mapping-theorem) 

  ##  [Open Mapping Theorem](https://puremath.quantumdictionary.io/open-mapping-theorem-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/index.php/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

In Banach spaces: any surjective continuous linear operator between Banach spaces maps open sets to open sets; equivalently, a surjective bounded linear operator sends some neighborhood of zero in the domain to a neighborhood of zero in the codomain.