Open mapping theorem

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title: "Open mapping theorem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public topic_path: "uncategorized" source: "https://en.wikipedia.org/wiki/Open_mapping_theorem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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.

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