In mathematical analysis, Littlewood's 4/3 inequality, named after John Edensor Littlewood, [1] is an inequality that holds for every complex-valued bilinear form defined on , the Banach space of scalar sequences that converge to zero.
Precisely, let or be a bilinear form. Then the following holds:
where
The exponent 4/3 is optimal, i.e., cannot be improved by a smaller exponent. [2] It is also known that for real scalars the aforementioned constant is sharp. [3]
Bohnenblust–Hille inequality [4] is a multilinear extension of Littlewood's inequality that states that for all -linear mapping the following holds:
In mathematics, a product is the result of multiplication, or an expression that identifies objects to be multiplied, called factors. For example, 21 is the product of 3 and 7, and is the product of and .
The Cauchy–Schwarz inequality is considered one of the most important and widely used inequalities in mathematics.
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz.
In mathematics, an infinite series of numbers is said to converge absolutely if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to converge absolutely if for some real number Similarly, an improper integral of a function, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces.
In abstract algebra and multilinear algebra, a multilinear form on a vector space over a field is a map
In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to J. H. C. Whitehead (1935) discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper Whitehead where he incorrectly claimed that no such manifold exists.
In mathematics, the real coordinate space of dimension n, denoted Rn or , is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers. Special cases are called the real lineR1 and the real coordinate planeR2. With component-wise addition and scalar multiplication, it is a real vector space, and its elements are called coordinate vectors.
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set which can be "inflated" or "scaled up" to eventually always include any given point of the vector space. Alternative terms are radial or absorbent set. Every neighborhood of the origin in every topological vector space is an absorbing subset.
In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced, in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set.
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or conditions are no longer required to hold absolutely and has instead weak solutions only with respect to certain "test vectors" or "test functions". In a strong formulation, the solution space is constructed such that these equations or conditions are already fulfilled.
In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let X1, ..., Xn be independent Bernoulli random variables taking values +1 and −1 with probability 1/2, then for every positive ,
In mathematics, the Grothendieck inequality states that there is a universal constant with the following property. If Mij is an n × n matrix with
In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the Lp spaces. Like the Lp spaces, they are Banach spaces. The spaces are named for Władysław Orlicz, who was the first to define them in 1932.
In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If f is an analytic function on an interval [a,b] ⊂ R, and at some point f and all of its derivatives are zero, then f is identically zero on all of [a,b]. Quasi-analytic classes are broader classes of functions for which this statement still holds true.
The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered by William Henry Young (1913) and extended by Hausdorff (1923). It is now typically understood as a rather direct corollary of the Plancherel theorem, found in 1910, in combination with the Riesz-Thorin theorem, originally discovered by Marcel Riesz in 1927. With this machinery, it readily admits several generalizations, including to multidimensional Fourier series and to the Fourier transform on the real line, Euclidean spaces, as well as more general spaces. With these extensions, it is one of the best-known results of Fourier analysis, appearing in nearly every introductory graduate-level textbook on the subject.
In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of
In mathematics, Katugampola fractional operators are integral operators that generalize the Riemann–Liouville and the Hadamard fractional operators into a unique form. The Katugampola fractional integral generalizes both the Riemann–Liouville fractional integral and the Hadamard fractional integral into a single form and It is also closely related to the Erdelyi–Kober operator that generalizes the Riemann–Liouville fractional integral. Katugampola fractional derivative has been defined using the Katugampola fractional integral and as with any other fractional differential operator, it also extends the possibility of taking real number powers or complex number powers of the integral and differential operators.
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