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 ${\displaystyle U\subseteq \mathbb {R} ^{n}}$ that have compact support. The space of all test functions, denoted by ${\displaystyle C_{c}^{\infty }(U),}$ is endowed with a certain topology, called the canonical LF-topology, that makes ${\displaystyle C_{c}^{\infty }(U)}$ into a complete Hausdorff locally convex TVS. The strong dual space of ${\displaystyle C_{c}^{\infty }(U)}$ is called the space of distributions on ${\displaystyle U}$ and is denoted by ${\displaystyle {\mathcal {D}}^{\prime }(U):=\left(C_{c}^{\infty }(U)\right)_{b}^{\prime },}$ where the "${\displaystyle b}$" subscript indicates that the continuous dual space of ${\displaystyle C_{c}^{\infty }(U),}$ denote by ${\displaystyle \left(C_{c}^{\infty }(U)\right)^{\prime },}$ 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 ${\displaystyle U=\mathbb {R} ^{n}}$ then the use of Schwartz functions^{[note 1]} as test functions gives rise to a certain subspace of ${\displaystyle {\mathcal {D}}^{\prime }(U)}$ whose elements are called tempered distributions. 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 ${\displaystyle {\mathcal {D}}^{\prime }(U)}$ 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 not subsets of ${\displaystyle C_{c}^{\infty }(U),}$ 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.^{[note 2]} Use of analytic test functions leads to Sato's theory of hyperfunctions.