Open mapping theorem may refer to:
*
Open mapping theorem (functional analysis)
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear o ...
(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
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures t ...
), states that a
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
continuous
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
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
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, the proof in the setting of topological groups uses the
Baire category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that th ...
.
See also
* In
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
, part of the
inverse function theorem
In mathematics, the inverse function theorem is a theorem that asserts that, if a real function ''f'' has a continuous derivative near a point where its derivative is nonzero, then, near this point, ''f'' has an inverse function. The inverse fu ...
which states that a continuously
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
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
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
''U'' ⊂ R
''n'' to R
''m'' is such that the
Jacobian derivative ''dF''(''x'') is
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
at every point ''x'' ∈ ''U'', then ''F'' is an open mapping.
* The
invariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n.
It states:
:If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeomor ...
theorem shows that certain mappings between subsets of R
''n'' are open.
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