Open mapping theorem
Encyclopedia
Open mapping theorem may refer to:
- Open mapping theorem (functional analysis)Open mapping theorem (functional analysis)In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem , is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map...
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 - Open mapping theorem (complex analysis)Open mapping theorem (complex analysis)In complex analysis, the open mapping theorem states that if U is a connected open subset of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map .The open mapping theorem points to the sharp difference between holomorphy and real-differentiability...
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.