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EXTREMELY IMPORTANT!
https://en.wikipedia.org/wiki/List_of_theorems
This is a list of notable theorems. Lists of theorems and similar statements include:
List of algebras
List of algorithms
List of axioms
List of conjectures
List of data structures
List of derivatives and integrals in alternative calculi
List of equations
List of fundamental theorems
List of hypotheses
List of inequalities
List of integrals
List of laws
List of lemmas
List of limits
List of logarithmic identities
List of mathematical functions
List of mathematical identities
List of mathematical proofs
List of misnamed theorems
List of scientific laws
List of theories
Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields.
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Inequalities in pure mathematics
Analysis
Agmon
In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon,[1] consist of two closely related interpolation inequalities between the Lebesgue space L∞ and the Sobolev spaces Hs. It is useful in the study of partial differential equations.
For the n-dimensional case, choose s1 and s2 such that s1 < n/2 < s2.
Then, if 0 < θ < 1 and n/2 = θ*s1 + (1 - θ)*s2, the following inequality holds for any u ∈ H(s2(Ω) :
‖u‖subSup(L∞(Ω), ) ≤ C*‖u‖subSup(Hs1(Ω), θ) * ‖u‖(H(s2(Ω), (1-θ)))
Askey–Gasper
Babenko–Beckner
Bernoulli
Bernstein (mathematical analysis)
Bessel
Bihari–LaSalle
Bohnenblust–Hille
Borell–Brascamp–Lieb
Brezis–Gallouet
Carleman
Chebyshev–Markov–Stieltjes inequalities
Chebyshev sum
Clarkson inequalities
Eilenberg
Fekete–Szegő
Fenchel
Friedrichs
Gagliardo–Nirenberg interpolation
In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the Lp-norms of different weak derivatives of a function through an interpolation inequality. The theorem is of particular importance in the framework of elliptic partial differential equations and was originally formulated by Emilio Gagliardo and Louis Nirenberg in 1958. The Gagliardo-Nirenberg inequality has found numerous applications in the investigation of nonlinear partial differential equations, and has been generalized to fractional Sobolev spaces by Haim Brezis and Petru Mironescu in the late 2010s.
Let 1 ≤ q ≤ + ∞ be a positive extended real quantity. Let j and m be non-negative integers such that j < m . Furthermore, let 1 ≤ r ≤ + ∞ be a positive extended real quantity, p ≥ 1 be real and θ ∈ [ 0 , 1 ] such that the relations
1 p = j n + θ ( 1 r − m n ) + 1 − θ q , j m ≤ θ ≤ 1
hold. Then,
‖ D j u ‖ L p ( R n ) ≤ C ‖ D m u ‖ L r ( R n ) θ ‖ u ‖ L q ( R n ) 1 − θ
for any u ∈ L q ( R n ) such that D m u ∈ L r ( R n ) , with two exceptional cases:
Gårding
Grothendieck
Grunsky inequalities
Hanner inequalities
Hardy
Hardy–Littlewood
Hardy–Littlewood–Sobolev
Harnack
Hausdorff–Young
Hermite–Hadamard
Hilbert
Hölder
Jackson
Jensen
Khabibullin conjecture on integral inequalities
Kantorovich
Karamata
Korn
Ladyzhenskaya
In mathematics, Ladyzhenskaya's inequality is any of a number of related functional inequalities named after the Soviet Russian mathematician Olga Aleksandrovna Ladyzhenskaya. The original such inequality, for functions of two real variables, was introduced by Ladyzhenskaya in 1958 to prove the existence and uniqueness of long-time solutions to the Navier–Stokes equations in two spatial dimensions (for smooth enough initial data).
Ladyzhenskaya's theorems are special cases of the Gagliardo–Nirenberg interpolation
Landau–Kolmogorov
Lebedev–Milin
Lieb–Thirring
Littlewood 4/3
Markov brothers'
Mashreghi–Ransford
Max–min
Minkowski
Poincaré
Popoviciu
Prékopa–Leindler
Rayleigh–Faber–Krahn
Remez
Riesz rearrangement
Schur test
Shapiro
Sobolev
Steffensen
Szegő
Three spheres
Trace inequalities
Trudinger theorem
Turán inequalities
Von Neumann
Wirtinger for functions
Young convolution
Young for products
Inequalities relating to means
Hardy–Littlewood maximal
Inequality of arithmetic and geometric means
Ky Fan
Levinson
Maclaurin
Mahler
Muirhead
Newton inequalities
Stein–Strömberg theorem
Combinatorics
Binomial coefficient bounds
Factorial bounds
XYZ
Fisher
Ingleton
Lubell–Yamamoto–Meshalkin
Nesbitt
Rearrangement
Schur
Shapiro
Stirling formula (bounds)
Differential equations
Grönwall
Geometry
See also: List of triangle inequalities
Alexandrov–Fenchel
Aristarchus
Barrow
Berger–Kazdan comparison theorem
Blaschke–Lebesgue
Blaschke–Santaló
Bishop–Gromov
Bogomolov–Miyaoka–Yau
Bonnesen
Brascamp–Lieb
Brunn–Minkowski
Castelnuovo–Severi
Cheng eigenvalue comparison theorem
Clifford theorem on special divisors
Cohn-Vossen
Erdős–Mordell
Euler theorem in geometry
Gromov for complex projective space
Gromov systolic for essential manifolds
Hadamard
Hadwiger–Finsler
Hinge theorem
Hitchin–Thorpe
Isoperimetric
Jordan
Jung theorem
Loewner torus
Łojasiewicz
Loomis–Whitney
Melchior
Milman reverse Brunn–Minkowski
Milnor–Wood
Minkowski first for convex bodies
Myers theorem
Noether
Ono
Pedoe
Ptolemy
Pu
Riemannian Penrose
Toponogov theorem
Triangle
Weitzenböck
Wirtinger (2-forms)
Information theory
Inequalities in information theory
Kraft
Log sum
Welch bounds
Algebra
Abhyankar
Pisier–Ringrose
Linear algebra
Abel
Bregman–Minc
Cauchy–Schwarz
Golden–Thompson
Hadamard
Hoffman-Wielandt
Peetre
Sylvester rank
Triangle
Trace inequalities
Eigenvalue inequalities
Bendixson
Weyl in matrix theory
Cauchy interlacing theorem
Poincaré separation theorem
Number theory
Bonse
Large sieve
Pólya–Vinogradov
Turán–Kubilius
Weyl
Probability theory and statistics
Azuma
Bennett, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount
Bhatia–Davis, an upper bound on the variance of any bounded probability distribution
Bernstein inequalities (probability theory)
Boole
Borell–TIS
BRS-inequality
Burkholder
Burkholder–Davis–Gundy inequalities
Cantelli
Chebyshev
Chernoff
Chung–Erdős
Concentration
Cramér–Rao
Doob martingale
Dvoretzky–Kiefer–Wolfowitz
Eaton, a bound on the largest absolute value of a linear combination of bounded random variables
Emery
Entropy power
Etemadi
Fannes–Audenaert
Fano
Fefferman
Fréchet inequalities
Gauss
Gauss–Markov theorem, the statement that the least-squares estimators in certain linear models are the best linear unbiased estimators
Gaussian correlation
Gaussian isoperimetric
Gibbs
Hoeffding
Hoeffding lemma
Jensen
Khintchine
Kolmogorov
Kunita–Watanabe
Le Cam theorem
Lenglart
Marcinkiewicz–Zygmund
Markov
McDiarmid
Paley–Zygmund
Pinsker
Popoviciu on variances
Prophet
Rao–Blackwell theorem
Ross conjecture, a lower bound on the average waiting time in certain queues
Samuelson
Shearer
Stochastic Gronwall
Talagrand concentration
Vitale random Brunn–Minkowski
Vysochanskiï–Petunin
Topology
Berger for Einstein manifolds
Inequalities particular to physics
Ahlswede–Daykin
Bell – see Bell theorem
Bell original
CHSH
Clausius–Duhem
Correlation – any of several inequalities
FKG
Ginibre inequal
ity
Griffiths
Heisenberg
Holley
Leggett–Garg
Riemannian Penrose
Rushbrooke
Tsirelson
See also
Comparison theorem
List of mathematical identities
Lists of mathematics topics
List of set identities and relations
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https://en.wikipedia.org/wiki/Schur_test
In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the L 2 → L 2 {\displaystyle L^{2}\to L^{2}} operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem).
Here is one version.[1] Let X , Y {\displaystyle X,\,Y} be two measurable spaces (such as R n {\displaystyle \mathbb {R} ^{n}}). Let T {\displaystyle \,T} be an integral operator with the non-negative Schwartz kernel K ( x , y ) {\displaystyle \,K(x,y)}, x ∈ X {\displaystyle x\in X}, y ∈ Y {\displaystyle y\in Y}:
T f ( x ) = ∫ Y K ( x , y ) f ( y ) d y . {\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy.}
If there exist real functions p ( x ) > 0 {\displaystyle \,p(x)>0} and q ( y ) > 0 {\displaystyle \,q(y)>0} and numbers α , β > 0 {\displaystyle \,\alpha ,\beta >0} such that
( 1 ) ∫ Y K ( x , y ) q ( y ) d y ≤ α p ( x ) {\displaystyle (1)\qquad \int _{Y}K(x,y)q(y)\,dy\leq \alpha p(x)}
for almost all x {\displaystyle \,x} and
( 2 ) ∫ X p ( x ) K ( x , y ) d x ≤ β q ( y ) {\displaystyle (2)\qquad \int _{X}p(x)K(x,y)\,dx\leq \beta q(y)}
for almost all y {\displaystyle \,y}, then T {\displaystyle \,T} extends to a continuous operator T : L 2 → L 2 {\displaystyle T:L^{2}\to L^{2}} with the operator norm
‖ T ‖ L 2 → L 2 ≤ α β . {\displaystyle \Vert T\Vert _{L^{2}\to L^{2}}\leq {\sqrt {\alpha \beta }}.}
Such functions p ( x ) {\displaystyle \,p(x)}, q ( y ) {\displaystyle \,q(y)} are called the Schur test functions.
In the original version, T {\displaystyle \,T} is a matrix and α = β = 1 {\displaystyle \,\alpha =\beta =1}.[2]
Common usage and Young's inequality
A common usage of the Schur test is to take p ( x ) = q ( y ) = 1. {\displaystyle \,p(x)=q(y)=1.} Then we get:
‖ T ‖ L 2 → L 2 2 ≤ sup x ∈ X ∫ Y | K ( x , y ) | d y ⋅ sup y ∈ Y ∫ X | K ( x , y ) | d x . {\displaystyle \Vert T\Vert _{L^{2}\to L^{2}}^{2}\leq \sup _{x\in X}\int _{Y}|K(x,y)|\,dy\cdot \sup _{y\in Y}\int _{X}|K(x,y)|\,dx.}
This inequality is valid no matter whether the Schwartz kernel K ( x , y ) {\displaystyle \,K(x,y)} is non-negative or not.
A similar statement about L p → L q {\displaystyle L^{p}\to L^{q}} operator norms is known as Young's inequality for integral operators:[3]
if
sup x ( ∫ Y | K ( x , y ) | r d y ) 1 / r + sup y ( ∫ X | K ( x , y ) | r d x ) 1 / r ≤ C , {\displaystyle \sup _{x}{\Big (}\int _{Y}|K(x,y)|^{r}\,dy{\Big )}^{1/r}+\sup _{y}{\Big (}\int _{X}|K(x,y)|^{r}\,dx{\Big )}^{1/r}\leq C,}
where r {\displaystyle r} satisfies 1 r = 1 - ( 1 p - 1 q ) {\displaystyle {\frac {1}{r}}=1-{\Big (}{\frac {1}{p}}-{\frac {1}{q}}{\Big )}}, for some 1 ≤ p ≤ q ≤ ∞ {\displaystyle 1\leq p\leq q\leq \infty }, then the operator T f ( x ) = ∫ Y K ( x , y ) f ( y ) d y {\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy} extends to a continuous operator T : L p ( Y ) → L q ( X ) {\displaystyle T:L^{p}(Y)\to L^{q}(X)}, with ‖ T ‖ L p → L q ≤ C . {\displaystyle \Vert T\Vert _{L^{p}\to L^{q}}\leq C.}
Proof is provided.
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