Spaces of test functions and distributions

In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex-valued (or sometimes real-valued) functions on a non-empty open subset that have compact support. The space of all test functions, denoted by is endowed with a certain topology, called the , that makes into a complete Hausdorff locally convex TVS. The strong dual space of is called and is denoted by where the "" subscript indicates that the continuous dual space of denoted by is endowed with the strong dual topology. There are other possible choices for the space of test functions, which lead to other different spaces of distributions. If then the use of Schwartz functions as test functions gives rise to a certain subspace of whose elements are called . These are important because they allow the Fourier transform to be extended from "standard functions" to tempered distributions. The set of tempered distributions forms a vector subspace of the space of distributions and is thus one example of a space of distributions; there are many other spaces of distributions. There also exist other major classes of test functions that are subsets of such as spaces of analytic test functions, which produce very different classes of distributions. The theory of such distributions has a different character from the previous one because there are no analytic functions with non-empty compact support. Use of analytic test functions leads to Sato's theory of hyperfunctions. The following notation will be used throughout this article: is a fixed positive integer and is a fixed non-empty open subset of Euclidean space denotes the natural numbers. will denote a non-negative integer or If is a function then will denote its domain and the of denoted by is defined to be the closure of the set in For two functions , the following notation defines a canonical pairing: A of size is an element in (given that is fixed, if the size of multi-indices is omitted then the size should be assumed to be ).