# Open mapping theorem

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Open_mapping_theorem
> Markdown URL: https://mediated.wiki/source/Open_mapping_theorem.md
> Source: https://en.wikipedia.org/wiki/Open_mapping_theorem
> Source revision: 1237532220
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

'''Open mapping theorem''' may refer to:

* [Open mapping theorem (functional analysis)](/source/Open_mapping_theorem_(functional_analysis)) (also known as the Banach–Schauder theorem), states that a surjective continuous linear transformation of a Banach space ''X'' onto a Banach space ''Y'' is an open mapping
* [Open mapping theorem (complex analysis)](/source/Open_mapping_theorem_(complex_analysis)), states that a non-constant holomorphic function on a connected open set in the complex plane is an open mapping
* Open mapping theorem ([topological groups](/source/topological_groups)), states that a [surjective](/source/surjective) continuous [homomorphism](/source/homomorphism) of a locally compact [Hausdorff](/source/Hausdorff_space) group ''G'' onto a locally compact Hausdorff group ''H'' is an open mapping if ''G'' is ''σ''-compact. Like the open mapping theorem in [functional analysis](/source/functional_analysis), the proof in the setting of topological groups uses the [Baire category theorem](/source/Baire_category_theorem).

== See also ==
* In [calculus](/source/calculus), part of the [inverse function theorem](/source/inverse_function_theorem) which states that a continuously [differentiable](/source/derivative) function between [Euclidean spaces](/source/Euclidean_spaces) whose [derivative matrix](/source/Jacobian_matrix_and_determinant) is invertible at a point is an open mapping in a neighborhood of the point.  More generally, if a mapping ''F''&nbsp;:&nbsp;''U''&nbsp;→&nbsp;'''R'''<sup>''m''</sup> from an [open set](/source/open_set) ''U''&nbsp;&sub;&nbsp;'''R'''<sup>''n''</sup> to '''R'''<sup>''m''</sup> is such that the [Jacobian](/source/Jacobian) derivative ''dF''(''x'') is [surjective](/source/surjective) at every point ''x''&nbsp;∈&nbsp;''U'', then ''F'' is an open mapping.
* The [invariance of domain](/source/invariance_of_domain) theorem shows that certain mappings between subsets of '''R'''<sup>''n''</sup> are open.

{{Set index article}}

---
Adapted from the Wikipedia article [Open mapping theorem](https://en.wikipedia.org/wiki/Open_mapping_theorem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Open_mapping_theorem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.