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# Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces. [1]

127 relations: Absolute continuity, Absolute convergence, Absolute value, Abuse of notation, Almost everywhere, Axiom of dependent choice, Ba space, Banach space, Birnbaum–Orlicz space, Bochner space, Borel set, Bounded function, Bounded operator, C*-algebra, Cauchy–Schwarz inequality, Central tendency, Chebyshev distance, Clarkson's inequalities, Closed graph theorem, Commutative property, Complete metric space, Complex number, Compressed sensing, Computational science, Computer science, Convergence in measure, Convergence of random variables, Counting measure, Cumulative distribution function, David Donoho, Density on a manifold, Elastic net regularization, Essential supremum and essential infimum, F-space, Fourier series, Fourier transform, Frigyes Riesz, Function (mathematics), Function space, Functional analysis, Generalized mean, Hahn–Banach theorem, Hardy space, Hardy–Littlewood maximal function, Harmonic analysis, Harmonic series (mathematics), Hausdorff–Young inequality, Hölder condition, Hölder's inequality, Henri Lebesgue, ... Expand index (77 more) »

## Absolute continuity

In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity.

## Absolute convergence

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.

## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

## Abuse of notation

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition (while being unlikely to introduce errors or cause confusion).

## Almost everywhere

In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities.

## Axiom of dependent choice

In mathematics, the axiom of dependent choice, denoted by \mathsf, is a weak form of the axiom of choice (\mathsf) that is still sufficient to develop most of real analysis.

## Ba space

In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma.

## Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

## Birnbaum–Orlicz space

In the mathematical analysis, and especially in real and harmonic analysis, a Birnbaum–Orlicz space is a type of function space which generalizes the L''p'' spaces.

## Bochner space

In mathematics, Bochner spaces are a generalization of the concept of ''Lp'' spaces to functions whose values lie in a Banach space which is not necessarily the space R or C of real or complex numbers.

## Borel set

In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.

## Bounded function

In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded.

## Bounded operator

In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).

## C*-algebra

C∗-algebras (pronounced "C-star") are an area of research in functional analysis, a branch of mathematics.

## Cauchy–Schwarz inequality

In mathematics, the Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas.

## Central tendency

In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.

## Chebyshev distance

In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension.

## Clarkson's inequalities

In mathematics, Clarkson's inequalities, named after James A. Clarkson, are results in the theory of ''L''''p'' spaces.

## Closed graph theorem

In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs.

## Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

## Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Compressed sensing

Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems.

## Computational science

Computational science (also scientific computing or scientific computation (SC)) is a rapidly growing multidisciplinary field that uses advanced computing capabilities to understand and solve complex problems.

## Computer science

Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.

## Convergence in measure

Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability.

## Convergence of random variables

In probability theory, there exist several different notions of convergence of random variables.

## Counting measure

In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be: the number of elements in the subset if the subset has finitely many elements, and &infin; if the subset is infinite.

## Cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF, also cumulative density function) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. In the case of a continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

## David Donoho

David Leigh Donoho (born March 5, 1957) is a professor of statistics at Stanford University, where he is also the Anne T. and Robert M. Bass Professor in the Humanities and Sciences.

## Density on a manifold

In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner.

## Elastic net regularization

In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods.

## Essential supremum and essential infimum

In mathematics, the concepts of essential supremum and essential infimum are related to the notions of supremum and infimum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for all elements in a set, but rather almost everywhere, i.e., except on a set of measure zero.

## F-space

In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d: V × V → R so that.

## Fourier series

In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.

## Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

## Frigyes Riesz

Frigyes Riesz (Riesz Frigyes,; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Function space

In mathematics, a function space is a set of functions between two fixed sets.

## Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

## Generalized mean

In mathematics, generalized means are a family of functions for aggregating sets of numbers, that include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).

## Hahn–Banach theorem

In mathematics, the Hahn–Banach theorem is a central tool in functional analysis.

## Hardy space

In complex analysis, the Hardy spaces (or Hardy classes) Hp are certain spaces of holomorphic functions on the unit disk or upper half plane.

## Hardy–Littlewood maximal function

In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis.

## Harmonic analysis

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).

## Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series: Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are,,, etc., of the string's fundamental wavelength.

## Hausdorff–Young inequality

In mathematics, the Hausdorff−Young inequality bounds the ''L''''q''-norm of the Fourier coefficients of a periodic function for q ≥ 2.

## Hölder condition

In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α, such that for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces.

## Hölder's inequality

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.

## Henri Lebesgue

Henri Léon Lebesgue (June 28, 1875 – July 26, 1941) was a French mathematician most famous for his theory of integration, which was a generalization of the 17th century concept of integration—summing the area between an axis and the curve of a function defined for that axis.

## Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

## Hilbert transform

In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t).

## Homogeneous function

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

## Improper integral

In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number, \infty, -\infty, or in some instances as both endpoints approach limits.

## Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.

## Infimum and supremum

In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.

## Information theory

Information theory studies the quantification, storage, and communication of information.

## Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

## Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## Σ-finite measure

In mathematics, a positive (or signed) measure μ defined on a ''&sigma;''-algebra Σ of subsets of a set X is called finite if μ(X) is a finite real number (rather than ∞).

## Kernel (set theory)

In set theory, the kernel of a function f may be taken to be either.

## L-infinity

In mathematics, ℓ∞ and L∞ are two related vector spaces.

## Lasso (statistics)

In statistics and machine learning, lasso (least absolute shrinkage and selection operator) (also Lasso or LASSO) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the statistical model it produces.

## Lebesgue integration

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis.

## Local boundedness

In mathematics, a function is locally bounded if it is bounded around every point.

## Locally convex topological vector space

In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces.

## Locally integrable function

In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition.

## Lorentz space

In mathematical analysis, Lorentz spaces, introduced by George Lorentz in the 1950s, are generalisations of the more familiar L^p spaces.

## Lp sum

In mathematics, and specifically in functional analysis, the Lp sum of a family of Banach spaces is a way of turning a subset of the product set of the members of the family into a Banach space in its own right.

## Marcinkiewicz interpolation theorem

In mathematics, the Marcinkiewicz interpolation theorem, discovered by, is a result bounding the norms of non-linear operators acting on ''L''p spaces.

## Markov's inequality

In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## McGraw-Hill Education

McGraw-Hill Education (MHE) is a learning science company and one of the "big three" educational publishers that provides customized educational content, software, and services for pre-K through postgraduate education.

## Mean

In mathematics, mean has several different definitions depending on the context.

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## Measurable function

In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open.

## Measure space

A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes.

## Median

The median is the value separating the higher half of a data sample, a population, or a probability distribution, from the lower half.

## Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

## Metrization theorem

In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.

## Minkowski distance

The Minkowski distance is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance.

## Minkowski inequality

In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces.

## Muckenhoupt weights

In mathematics, the class of Muckenhoupt weights consists of those weights for which the Hardy–Littlewood maximal operator is bounded on.

## Multiplication operator

In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.

## Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

## Normed vector space

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined.

## Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

## Operator norm

In mathematics, the operator norm is a means to measure the "size" of certain linear operators.

## Paul Lévy (mathematician)

Paul Pierre Lévy (15 September 1886 – 15 December 1971) was a French mathematician who was active especially in probability theory, introducing fundamental concepts such as local time, stable distributions and characteristic functions.

## Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.

## Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

## Property of Baire

A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such that A \bigtriangleup U is meager (where \bigtriangleup denotes the symmetric difference).

## Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

## Quasinorm

In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by for some K > 0.

## Quotient space (topology)

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.

In mathematics, the Radon–Nikodym theorem is a result in measure theory.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Reflexive space

In the area of mathematics known as functional analysis, a reflexive space is a Banach space (or more generally a locally convex topological vector space) that coincides with the continuous dual of its continuous dual space, both as linear space and as topological space.

## Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.

## Riesz–Fischer theorem

In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space ''L''2 of square integrable functions.

## Riesz–Thorin theorem

In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem is a result about interpolation of operators.

## Root mean square

In statistics and its applications, the root mean square (abbreviated RMS or rms) is defined as the square root of the mean square (the arithmetic mean of the squares of a set of numbers).

## Saharon Shelah

Saharon Shelah (שהרן שלח) is an Israeli mathematician.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

## Signal processing

Signal processing concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements.

## Singular integral

In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations.

## Square-integrable function

In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.

## Standard deviation

In statistics, the standard deviation (SD, also represented by the Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.

## Statistical dispersion

In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed.

## Statistics

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.

## Stefan Banach

Stefan Banach (30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the world's most important and influential 20th-century mathematicians.

## Stochastic calculus

Stochastic calculus is a branch of mathematics that operates on stochastic processes.

In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element.

## Superellipse

A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape.

## Taxicab geometry

A taxicab geometry is a form of geometry in which the usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates.

## Tikhonov regularization

Tikhonov regularization, named for Andrey Tikhonov, is the most commonly used method of regularization of ill-posed problems.

## Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

## Topological vector space

In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.

## Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

## Uniformly convex space

In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## Von Neumann algebra

In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator.

## Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.

## Zero to the power of zero

Zero to the power of zero, denoted by 00, is a mathematical expression with no obvious value.

## References

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