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In mathematics, inverse mapping theorem may refer to: the inverse function theorem on the existence of local inverses for functions with non-singular derivatives the bounded inverse theorem on the boundedness of the inverse for invertible bounded linear operators on Banach spaces
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
The inverse function theorem can be generalized to functions of several variables. ... [0, ∞) denote the squaring map, such that f(x) = x 2 for all x in R, ...
From the establishment of the inverse function theorem, the following mapping can be defined. For the domain U , V of the n -dimensional complex space C n {\displaystyle \mathbb {C} ^{n}} , the bijective holomorphic function ϕ : U → V {\displaystyle \phi :U\to V} and the inverse mapping ϕ − 1 : V → U {\displaystyle \phi ^{-1}:V\to U} is ...
The open mapping theorem has several important consequences: If : is a bijective continuous linear operator between the Banach spaces and , then the inverse operator: is continuous as well (this is called the bounded inverse theorem). [10]
The map f is a submersion at a point ... The theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure).
A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map. The inverse function theorem implies that a smooth map : is a local diffeomorphism if and only if the derivative: is a linear isomorphism for all points .
The open mapping theorem forces the inverse function (defined on the image of ) to be holomorphic. Thus, under this definition, a map is conformal if and only if it is biholomorphic. The two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative.