The main motivation behind distribution theory is to extend the common linear operators on functions, such as the derivativePlanetmathPlanetmath, convolution, and the Fourier transformMathworldPlanetmath, so that they also apply to the singular, non-smooth, or non-integrable functions that regularly appear in both theoretical and applied analysisMathworldPlanetmath.

Distribution theory also seeks to define suitable structuresMathworldPlanetmath on the spaces of functions involved to ensure the convergence of suitable approximating functions, and the continuity of certain operatorsMathworldPlanetmath. For example, the limit of derivatives should be equal to the derivative of the limit, with some definition of the limiting operationMathworldPlanetmath.

When this program is carried out, inevitably we find that we have to enlarge the space of objects that we would consider as “functions”. For example, the derivative of a step functionPlanetmathPlanetmath is the Dirac delta function with a spike at the discontinuousMathworldPlanetmath step; the Fourier transform of a constant function is also a Dirac delta function, with the spike representing infiniteMathworldPlanetmath spectral magnitude at one single frequency. (These facts, of course, had long been used in engineering mathematics.)

Remark: Dirac’s notion of delta distributions was introduced to facilitate computations in Quantum Mechanics, however without having at the beginning a proper mathematical definition. In part as a (negative) reaction to such a state of affairs, von Neumann produced a mathematically well-defined foundation of Quantum Mechanics (http://planetmath.org/QuantumGroupsAndVonNeumannAlgebras) based on actions of self-adjoint operators on Hilbert spacesMathworldPlanetmath which is still currently in use, with several significant additions such as Frechét nuclear spaces and quantum groupsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

There are several theories of such generalized functions. In this entry, we describe Schwartz’ theory of distributions, which is probably the most widely used.

Essentially, a distribution on is a linear mapping that takes a smooth functionMathworldPlanetmath (with compact support) on into a real number. For example, the delta distribution is the map,


while any smooth function g on induces a distribution


Distributions are also well behaved under coordinate changes, and can be defined onto a manifold. Differential forms with distribution valued coefficients are called currents. However, it is not possible to define a productPlanetmathPlanetmath of two distributions generalizing the product of usual functions.

Formal definition

A note on notation. In distribution theory, the topological vector spaceMathworldPlanetmath of smooth functions with compact support on an open set Un is traditionally denoted by 𝒟(U). Let us also denote by 𝒟K(U) the subset of 𝒟(U) of functions with supportMathworldPlanetmath in a compact set KU.

Definition 1 (Distribution).

A distribution is a linear continuous functional on D(U), i.e., a linear continuous mapping D(U)C. The set of all distributions on U is denoted by D(U).

Suppose T is a linear functionalMathworldPlanetmath on 𝒟(U). Then T is continuousMathworldPlanetmath if and only if T is continuous in the origin (see this page (http://planetmath.org/ContinuousLinearMapping)). This condition can be rewritten in various ways, and the below theorem gives two convenient conditions that can be used to prove that a linear mapping is a distribution.

Theorem 1.

Let U be an open set in Rn, and let T be a linear functional on D(U). Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    T is a distribution.

  2. 2.

    If K is a compact set in U, and {ui}i=1 be a sequence in 𝒟K(U), such that for any multi-index α, we have


    in the supremum normMathworldPlanetmath as i, then T(ui)0 in .

  3. 3.

    For any compact set K in U, there are constants C>0 and k{1,2,} such that for all u𝒟K(U), we have

    |T(u)| C|α|k||Dαu||, (1)

    where α is a multi-index, and |||| is the supremum norm.

Proof The equivalence of (2) and (3) can be found on this page (http://planetmath.org/EquivalenceOfConditions2And3), and the equivalence of (1) and (3) is shown in [1].

Distributions of order k

If T is a distribution on an open set U, and the same k can be used for any K in the above inequalityMathworldPlanetmath, then T is a distribution of order k. The set of all such distributions is denoted by Dk(U).

Both usual functions and the delta distribution are of order 0. One can also show that by differentiating a distribution its order increases by at most one. Thus, in some sense, the order is a measure of how ”smooth” a distribution is.

Topology for 𝒟(U)

The standard topology for 𝒟(U) is the weak topologyMathworldPlanetmath. In this topology, a sequence {Ti}i=1 of distributions (in 𝒟(U)) convergesPlanetmathPlanetmath to a distribution T𝒟(U) if and only if

Ti(u)T(u)(in ) as i

for every u𝒟(U).


A common notation for the action of a distribution T onto a test function u𝒟(U) (i.e., T(u) with above notation) is T,u. The motivation for this comes from this example (http://planetmath.org/EveryLocallyIntegrableFunctionIsADistribution).


  • 1 W. Rudin, Functional AnalysisMathworldPlanetmath, McGraw-Hill Book Company, 1973.
  • 2 L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
Title distribution
Canonical name Distribution
Date of creation 2013-03-22 13:44:08
Last modified on 2013-03-22 13:44:08
Owner matte (1858)
Last modified by matte (1858)
Numerical id 23
Author matte (1858)
Entry type Definition
Classification msc 46-00
Classification msc 46F05
Synonym ‘generalized function’
Related topic ExampleOfDiracSequence
Related topic DiracDeltaFunction
Related topic DiscreteTimeFourierTransformInRelationWithItsContinousTimeFourierTransfrom
Related topic QuantumGroups
Related topic FourierStieltjesAlgebraOfAGroupoid
Related topic QuantumOperatorAlgebrasInQuantumFieldTheories
Related topic QFTOrQuantumFieldTheories
Related topic QuantumGroup
Defines distribution of finite order