# Category: Lemmas in analysis

Sard's theorem
In mathematics, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set o
Fundamental lemma of calculus of variations
In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point.Accordingly, the necessary condi
Lions–Magenes lemma
In mathematics, the Lions–Magenes lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a criterion for moving a time derivative of a functi
Riesz's lemma
Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense. The lemma may als
Itô's lemma
In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent fun
Pugh's closing lemma
In mathematics, Pugh's closing lemma is a result that links periodic orbit solutions of differential equations to chaotic behaviour. It can be formally stated as follows: Let be a diffeomorphism of a
Watson's lemma
In mathematics, Watson's lemma, proved by G. N. Watson (1918, p. 133), has significant application within the theory on the asymptotic behavior of integrals.
Cotlar–Stein lemma
In mathematics, in the field of functional analysis, the Cotlar–Stein almost orthogonality lemma is named after mathematicians Mischa Cotlarand Elias Stein. It may be used to obtain information on the
Stechkin's lemma
In mathematics – more specifically, in functional analysis and numerical analysis – Stechkin's lemma is a result about the ℓq norm of the tail of a sequence, when the whole sequence is known to have f
Lebesgue's lemma
For Lebesgue's lemma for open covers of compact spaces in topology see Lebesgue's number lemma In mathematics, Lebesgue's lemma is an important statement in approximation theory. It provides a bound f
Fatou's lemma
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma
Aubin–Lions lemma
In mathematics, the Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful in the study of
Rising sun lemma
In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz, used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zy
Jordan's lemma
In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. The lemma is named after the French mathem
Calderón–Zygmund lemma
In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmun
Ehrling's lemma
In mathematics, Ehrling's lemma, also known as Lions' lemma, is a result concerning Banach spaces. It is often used in functional analysis to demonstrate the equivalence of certain norms on Sobolev sp
Halanay inequality
Halanay inequality is a comparison theorem for differential equations with delay. This inequality and its generalizations have been applied to analyze the stability of delayed differential equations,
Borel's lemma
In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations.
Wiener's lemma
In mathematics, Wiener's lemma is a well-known identity which relates the asymptotic behaviour of the Fourier coefficients of a Borel measure on the circle to its atomic part. This result admits an an
Céa's lemma
Céa's lemma is a lemma in mathematics. Introduced by Jean Céa in his Ph.D. dissertation, it is an important tool for proving error estimates for the finite element method applied to elliptic partial d
Schwarz lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than
Auerbach's lemma
In mathematics, Auerbach's lemma, named after Herman Auerbach, is a theorem in functional analysis which asserts that a certain property of Euclidean spaces holds for general finite-dimensional normed
Stewart–Walker lemma
The Stewart–Walker lemma provides necessary and sufficient conditions for the linear perturbation of a tensor field to be gauge-invariant. if and only if one of the following holds 1. 2. is a constant
Estimation lemma
In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral. If f is a complex-valued, continuous function on the contour Γ and if its absolute va
Mazur's lemma
In mathematics, Mazur's lemma is a result in the theory of Banach spaces. It shows that any weakly convergent sequence in a Banach space has a sequence of convex combinations of its members that conve
Oka's lemma
In mathematics, Oka's lemma, proved by Kiyoshi Oka, states that in a domain of holomorphy in , the function is plurisubharmonic, where is the distance to the boundary. This property shows that the dom
Spijker's lemma
In mathematics, Spijker's lemma is a result in the theory of rational mappings of the Riemann sphere. It states that the image of a circle under a complex rational map with numerator and denominator h
Closed and exact differential forms
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, th
Morse–Palais lemma
In mathematics, the Morse–Palais lemma is a result in the calculus of variations and theory of Hilbert spaces. Roughly speaking, it states that a smooth enough function near a critical point can be ex
Riemann–Lebesgue lemma
In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L1 function vanishes at infinity. It is of imp
Grönwall's inequality
In mathematics, Grönwall's inequality (also called Grönwall's lemma or the Grönwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or by the solutio
Bramble–Hilbert lemma
In mathematics, particularly numerical analysis, the Bramble–Hilbert lemma, named after James H. Bramble and Stephen Hilbert, bounds the error of an approximation of a function by a polynomial of orde
Malliavin's absolute continuity lemma
In mathematics — specifically, in measure theory — Malliavin's absolute continuity lemma is a result due to the French mathematician Paul Malliavin that plays a foundational rôle in the regularity (sm
Weyl's lemma (Laplace equation)
In mathematics, Weyl's lemma, named after Hermann Weyl, states that every weak solution of Laplace's equation is a smooth solution. This contrasts with the wave equation, for example, which has weak s