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.
The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
x
OK