Open mapping theorem

In mathematics, there are two theorems with the name "open mapping theorem". In both cases, they give conditions under which certain maps are open maps, i.e. they map open sets to open sets. They are significant results in their respective contexts since, unlike inverse images, direct images of functions are much less tractable in general.

• In functional analysis, the open mapping theorem or Banach–Schauder theorem states that a surjective continuous linear transformation of a Banach space X onto a Banach space Y is an open mapping.

• In complex analysis, the open mapping theorem states that a non-constant holomorphic function on a connected open set in the complex plane is an open mapping.

• 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.