continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function...

the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation)...

Mapping theorem may refer to Continuous mapping theorem, a statement regarding the stability of convergence under mappings Mapping theorem (point process)...

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f {\displaystyle...

In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz...

hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector...

Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees...

mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X {\displaystyle...

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)...

Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important...

Next we apply the continuous mapping theorem, recognizing the functions g(x,y) = x + y, g(x,y) = xy, and g(x,y) = x y−1 are continuous (for the last function...

notation Skorokhod's representation theorem The Tweedie convergence theorem Slutsky's theorem Continuous mapping theorem Bickel et al. 1998, A.8, page 475...

In complex analysis, the Riemann mapping theorem states that if U {\displaystyle U} is a non-empty simply connected open subset of the complex number...

Leray–Schauder theorem which was proved earlier by Juliusz Schauder and Jean Leray. The statement is as follows: Let f {\displaystyle f} be a continuous and compact...

Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, published...

and y in M. Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no...

In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of...

Rouché's theorem is to prove the open mapping theorem for analytic functions. We refer to the article for the proof. A stronger version of Rouché's theorem was...

Central limit theorem (probability) Clark–Ocone theorem (stochastic processes) Continuous mapping theorem (probability theory) Cramér's theorem (large deviations)...

conformal mappings are precisely the locally invertible complex analytic functions. In three and higher dimensions, Liouville's theorem sharply limits...

the contraction mapping theorem. For functions of a single variable, the theorem states that if f {\displaystyle f} is a continuously differentiable function...

. An example where this version of the theorem applies is given by the real-valued cube root function mapping x ↦ x 1 / 3 {\displaystyle x\mapsto x^{1/3}}...

the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariate continuous function f : [ 0 , 1 ] n → R {\displaystyle...

In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each...

In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in R...

Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard, Ernst Lindelöf...

points. Every continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively...

quasiconformal mappings in two dimensions is the measurable Riemann mapping theorem, proved by Lars Ahlfors and Lipman Bers. The theorem generalizes the...

Markov–Kakutani fixed-point theorem (1936-1938) and the Ryll-Nardzewski fixed-point theorem (1967) for continuous affine self-mappings of compact convex sets...

noted in Open mapping theorem (functional analysis) § Statement and proof, it is enough to prove the open mapping theorem for a continuous linear operator...