Get it on Google Play
New! Download Unionpedia on your Android™ device!
Faster access than browser!

Sobolev space

Index Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function itself and its derivatives up to a given order. [1]

67 relations: Absolute continuity, Academic Press, Algebra over a field, Almost everywhere, Banach space, Besov space, Bessel potential, Birkhäuser, Cantor function, Cocompact embedding, Compact operator, Complete metric space, Cone condition, Continuous function, Derivative, Differentiable function, Differential equation, Dirichlet boundary condition, Dirichlet integral, Dirichlet problem, Distribution (mathematics), Domain (mathematical analysis), Embedding, Fourier series, Friedrich Bessel, Function (mathematics), Graduate Studies in Mathematics, Hölder condition, Hilbert space, Inner product space, Integration by parts, Interpolation space, James Serrin, Laplace operator, Lebesgue integration, Lipschitz continuity, Lipschitz domain, Locally integrable function, Lp space, Mathematician, Mathematics, Multi-index notation, Multiplier (Fourier analysis), Nachman Aronszajn, Natural number, Normed vector space, Open set, Orthonormal basis, Otto M. Nikodym, Parseval's theorem, ..., Partial derivative, Partial differential equation, Poincaré inequality, Riesz potential, Separable space, Sergei Sobolev, Smoothness, Sobolev inequality, Spherical coordinate system, Springer Science+Business Media, Support (mathematics), Trace operator, Uniform norm, Unit sphere, Vector space, Weak derivative, World Scientific. Expand index (17 more) »

Absolute continuity

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

New!!: Sobolev space and Absolute continuity · See more »

Academic Press

Academic Press is an academic book publisher.

New!!: Sobolev space and Academic Press · See more »

Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.

New!!: Sobolev space and Algebra over a field · See more »

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.

New!!: Sobolev space and Almost everywhere · See more »

Banach space

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

New!!: Sobolev space and Banach space · See more »

Besov space

In mathematics, the Besov space (named after Oleg Vladimirovich Besov) B^s_(\mathbf) is a complete quasinormed space which is a Banach space when.

New!!: Sobolev space and Besov space · See more »

Bessel potential

In mathematics, the Bessel potential is a potential (named after Friedrich Wilhelm Bessel) similar to the Riesz potential but with better decay properties at infinity.

New!!: Sobolev space and Bessel potential · See more »


Birkhäuser is a former Swiss publisher founded in 1879 by Emil Birkhäuser.

New!!: Sobolev space and Birkhäuser · See more »

Cantor function

In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous.

New!!: Sobolev space and Cantor function · See more »

Cocompact embedding

In mathematics, cocompact embeddings are embeddings of normed vector spaces possessing a certain property similar to but weaker than compactness.

New!!: Sobolev space and Cocompact embedding · See more »

Compact operator

In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset (has compact closure) of Y. Such an operator is necessarily a bounded operator, and so continuous.

New!!: Sobolev space and Compact operator · See more »

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).

New!!: Sobolev space and Complete metric space · See more »

Cone condition

In mathematics, the cone condition is a property which may be satisfied by a subset of an Euclidean space.

New!!: Sobolev space and Cone condition · See more »

Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

New!!: Sobolev space and Continuous function · See more »


The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

New!!: Sobolev space and Derivative · See more »

Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

New!!: Sobolev space and Differentiable function · See more »

Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

New!!: Sobolev space and Differential equation · See more »

Dirichlet boundary condition

In mathematics, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859).

New!!: Sobolev space and Dirichlet boundary condition · See more »

Dirichlet integral

In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet.

New!!: Sobolev space and Dirichlet integral · See more »

Dirichlet problem

In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.

New!!: Sobolev space and Dirichlet problem · See more »

Distribution (mathematics)

Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.

New!!: Sobolev space and Distribution (mathematics) · See more »

Domain (mathematical analysis)

In mathematical analysis, a domain is any connected open subset of a finite-dimensional vector space.

New!!: Sobolev space and Domain (mathematical analysis) · See more »


In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

New!!: Sobolev space and Embedding · See more »

Fourier series

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

New!!: Sobolev space and Fourier series · See more »

Friedrich Bessel

Friedrich Wilhelm Bessel (22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist and geodesist.

New!!: Sobolev space and Friedrich Bessel · See more »

Function (mathematics)

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

New!!: Sobolev space and Function (mathematics) · See more »

Graduate Studies in Mathematics

Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS).

New!!: Sobolev space and Graduate Studies in Mathematics · See more »

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.

New!!: Sobolev space and Hölder condition · See more »

Hilbert space

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

New!!: Sobolev space and Hilbert space · See more »

Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

New!!: Sobolev space and Inner product space · See more »

Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative.

New!!: Sobolev space and Integration by parts · See more »

Interpolation space

In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach spaces.

New!!: Sobolev space and Interpolation space · See more »

James Serrin

James Burton Serrin (1 November 1926, Chicago, Illinois – 23 August 2012, Minneapolis, Minnesota) was an American mathematician, and a professor at University of Minnesota.

New!!: Sobolev space and James Serrin · See more »

Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.

New!!: Sobolev space and Laplace operator · See more »

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.

New!!: Sobolev space and Lebesgue integration · See more »

Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

New!!: Sobolev space and Lipschitz continuity · See more »

Lipschitz domain

In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function.

New!!: Sobolev space and Lipschitz domain · See more »

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.

New!!: Sobolev space and Locally integrable function · See more »

Lp space

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

New!!: Sobolev space and Lp space · See more »


A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

New!!: Sobolev space and Mathematician · See more »


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

New!!: Sobolev space and Mathematics · See more »

Multi-index notation

Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.

New!!: Sobolev space and Multi-index notation · See more »

Multiplier (Fourier analysis)

In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions.

New!!: Sobolev space and Multiplier (Fourier analysis) · See more »

Nachman Aronszajn

Nachman Aronszajn (26 July 1907 – 5 February 1980) was a Polish American mathematician.

New!!: Sobolev space and Nachman Aronszajn · See more »

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").

New!!: Sobolev space and Natural number · See more »

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.

New!!: Sobolev space and Normed vector space · See more »

Open set

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

New!!: Sobolev space and Open set · See more »

Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

New!!: Sobolev space and Orthonormal basis · See more »

Otto M. Nikodym

Otto Marcin Nikodym (3 August 1887 – 4 May 1974) was a Polish mathematician.

New!!: Sobolev space and Otto M. Nikodym · See more »

Parseval's theorem

In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.

New!!: Sobolev space and Parseval's theorem · See more »

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

New!!: Sobolev space and Partial derivative · See more »

Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

New!!: Sobolev space and Partial differential equation · See more »

Poincaré inequality

In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré.

New!!: Sobolev space and Poincaré inequality · See more »

Riesz potential

In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz.

New!!: Sobolev space and Riesz potential · See more »

Separable space

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

New!!: Sobolev space and Separable space · See more »

Sergei Sobolev

Sergei Lvovich Sobolev (Серге́й Льво́вич Со́болев; 6 October 1908 – 3 January 1989) was a Soviet mathematician working in mathematical analysis and partial differential equations.

New!!: Sobolev space and Sergei Sobolev · See more »


In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

New!!: Sobolev space and Smoothness · See more »

Sobolev inequality

In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces.

New!!: Sobolev space and Sobolev inequality · See more »

Spherical coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.

New!!: Sobolev space and Spherical coordinate system · See more »

Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

New!!: Sobolev space and Springer Science+Business Media · See more »

Support (mathematics)

In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.

New!!: Sobolev space and Support (mathematics) · See more »

Trace operator

In mathematics, the concept of trace operator plays an important role in studying the existence and uniqueness of solutions to boundary value problems, that is, to partial differential equations with prescribed boundary conditions.

New!!: Sobolev space and Trace operator · See more »

Uniform norm

In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number This norm is also called the supremum norm, the Chebyshev norm, or the infinity norm. The name "uniform norm" derives from the fact that a sequence of functions \ converges to f under the metric derived from the uniform norm if and only if f_n converges to f uniformly.

New!!: Sobolev space and Uniform norm · See more »

Unit sphere

In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point.

New!!: Sobolev space and Unit sphere · See more »

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.

New!!: Sobolev space and Vector space · See more »

Weak derivative

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space \mathrm^1().

New!!: Sobolev space and Weak derivative · See more »

World Scientific

World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore.

New!!: Sobolev space and World Scientific · See more »

Redirects here:

H1 norm, H1-norm, Sobelov space, Sobolev classes (of functions), Sobolev norm, Sobolev spaces, Sobolev theory, Sobolov space, W 2^1.


[1] https://en.wikipedia.org/wiki/Sobolev_space

Hey! We are on Facebook now! »