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.

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 ⊂ Rⁿ 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 Rⁿ are open.

See also