Open mapping theorem

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

* 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), 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), states that a surjective continuous homomorphism of a locally compact Hausdorff 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, the proof in the setting of topological groups uses the Baire category theorem.

== See also == * In calculus, part of the inverse function theorem which states that a continuously differentiable function between Euclidean spaces whose derivative matrix is invertible at a point is an open mapping in a neighborhood of the point. More generally, if a mapping ''F'' : ''U'' → '''R'''''m'' from an open set ''U'' ⊂ '''R'''''n'' to '''R'''''m'' is such that the Jacobian derivative ''dF''(''x'') is surjective at every point ''x'' ∈ ''U'', then ''F'' is an open mapping. * The invariance of domain theorem shows that certain mappings between subsets of '''R'''''n'' are open.