In

mathematical analysis,

distributions are

generalized

functions that permit the

derivative to be

extended to

things which are not necessarily

smooth. The

theory of

distributions was founded in the middle of the

20th century by the

French analyst Laurent Schwartz, and

independently by

I. M. Gelfand and the

Russian school. In

Russian they are still called

generalized

functions.

Distributions on a

manifold X are defined as

continuous linear functionals on the

space of

smooth functions having

compact support. In

distribution theory this

space is

conventionally denoted by

**D**(X) or C

^{∞}_{c}(X), and the

distributions therefore by

**D′**(X) or C

^{-∞}(X). Because

**D**(X) is not

metrizable (unless X is

compact) but only an

LF space, the

continuity condition is a little

technical to state. (An

LF space is the

inductive limit of an ascending sequence of

Fréchet spaces: roughly, the

completeness condition is as good as that of a

Fréchet space but you only have a

uniformity with which to state it rather than a

metric.)

The advantage of

distributions is that they can be

differentiated indefinitely even though they may not be "

smooth". We simply imagine that the

integration by parts formula is valid, and define the

derivative of a distribution u to be
u′(φ) = u(-&phi′). This notion of

differentiation turns out to yield a

calculus with the correct properties.

Note that the word *distribution* is used by statisticians to mean something very different.

For more information, consult a textbook on partial differential equations for pure mathematicians. I like the first volume of Michael Taylor's three-volume work.