# Generalized functions, space of

distribution space

The space dual to the space of test (sufficiently good) functions. The Fréchet–Schwartz spaces (cf. Fréchet space) (type FS) and the spaces strongly dual to them (type DFS) play an important role here. A space of type FS is the projective limit of a directed set of Banach spaces and its dual is a space of type DFS. A space of type DFS is the inductive limit of a directed set of Banach spaces and its dual is a space of type FS. Spaces of types FS and DFS are complete, separable, reflexive, and Montel. In spaces of types FS and DFS weak and strong convergence coincide.

## Contents

### Examples of spaces of test and generalized functions.

1) The spaces $S$ and $S ^ \prime$. The space $S = S ( \mathbf R ^ {n} )$ of (rapidly-decreasing) test functions consists of the $C ^ \infty ( \mathbf R ^ {n} )$-functions that together with all their derivatives decrease at infinity faster than any power of $| x | ^ {- 1}$. This space is the projective limit of the sequence of Banach spaces $S _ {p}$, $p = 0, 1, \dots$ consisting of the $C ^ {p} ( \mathbf R ^ {n} )$-functions with norm

$$\phi \rightarrow \| \phi \| _ {p} = \ \sup _ {\begin{array}{c} | \alpha | \leq p \\ x \end{array} } \ ( 1 + | x | ^ {2} ) ^ {p/2} | D ^ \alpha \phi ( x) | ,$$

and the inclusion $S _ {p+1} \subset S _ {p}$ is compact; $S$ is of type FS. The dual space $S ^ \prime = S ^ \prime ( \mathbf R ^ {n} )$ (the space of generalized functions of slow growth) is the inductive limit of the sequence of Banach spaces $S _ {p} ^ \prime$, where the imbedding $S _ {p} ^ \prime \subset S _ {p+1} ^ \prime$ is compact, so that $S ^ \prime$ is of type DFS. If a sequence of generalized functions is (weakly) convergent in $S ^ \prime$, then it converges with respect to the norm of functionals in some $S _ {p} ^ \prime$. The Fourier transformation is an isomorphism on the spaces $S$ and $S ^ \prime$.

2) The spaces $D ( O)$ and $D ^ \prime ( O)$ ($O$ an open set in $\mathbf R ^ {n}$). The space of test functions consists of the $C ^ \infty ( O)$-functions that have compact support in $O$ (see Support of a generalized function). It is endowed with the topology of the strong inductive limit of the (increasing) sequence of spaces $C _ {0} ^ \infty ( \overline{O} _ {k} ) ,$ $k = 1 , 2, \dots$ of type FS, where $\{ O _ {k} \}$ is a strictly-increasing sequence of open sets that exhausts $O$, $O _ {k} \subset \subset O _ {k+1}$, $\overline{O} _ {k}$ compact, $\cup _ {k} O _ {k} = O$. The space $C _ {0} ^ \infty ( \overline{O} _ {k} )$ is the projective limit of the (decreasing) sequence of Banach spaces $C _ {0} ^ {p} ( \overline{O} _ {k} )$, $p = 0 , 1, \dots$ consisting of the $C ^ {p} ( \mathbf R ^ {n} )$ functions with support in $\overline{O} _ {k}$ and with norm

$$\phi \rightarrow \| \phi \| _ {p} ^ \prime = \ \max _ {\begin{array}{c} | \alpha | \leq p \\ x \end{array} } \ | D ^ \alpha \phi ( x) | ,$$

where the imbedding $C _ {0} ^ {p+1} ( \overline{O} _ {k} ) \subset C _ {0} ^ {p} ( \overline{O} _ {k} )$ is compact. Let $D ^ \prime ( O)$ be the space (strongly) dual to $D ( O)$; $D = D ( \mathbf R ^ {n} )$ and $D ^ \prime = D ^ \prime ( \mathbf R ^ {n} )$. A sequence of test functions in $D ( O)$ converges in $D ( O)$ if it converges in some space $C _ {0} ^ \infty ( \overline{O} _ {k} )$. A sequence of generalized functions in $D ^ \prime ( O)$ converges in $D ^ \prime ( O)$ if it converges on every element of $D ( O)$ (weak convergence). For a linear functional $f$ on $D ( O)$ to be a generalized function in $D ^ \prime ( O)$ it is necessary and sufficient that for any open set $O ^ \prime \subset \subset O$ there exist numbers $K$ and $m$ such that

$$| ( f , \phi ) | \leq \ K \| \phi \| _ {m} ^ \prime ,\ \ \phi \in D ( O ^ \prime ) .$$

The space $D ^ \prime ( O)$ is (weakly) complete: If a sequence of generalized functions $f _ {k} \in D ^ \prime ( O)$, $k = 1 , 2, \dots$ is such that for any $\phi$ in $D ( O)$ the sequence of numbers $( f _ {k} , \phi )$ converges, then the functional

$$( f , \phi ) = \ \lim\limits _ {k \rightarrow \infty } \ ( f _ {k} , \phi )$$

belongs to $D ^ \prime ( O)$. A generalized function in $D ^ \prime ( O)$ has unrestricted "growth" in a neighbourhood of the boundary $\partial O$; in particular, any function $f \in L _ { \mathop{\rm loc} } ^ {1} ( O)$ determines a generalized function in $D ^ \prime ( O)$ by the formula

$$\phi \rightarrow ( f , \phi ) = \ \int\limits f ( x) \phi ( x) d x ,\ \ \phi \in D ( O) .$$

3) The spaces $\Phi$ and $\Phi ^ \prime$. Let $\Phi _ {p}$ be the Banach space of all functions $\phi ( z)$, $z = x + i y$, that are holomorphic in the tubular neighbourhood $| y | < \rho$, $x \in \mathbf R ^ {n}$, with norm

$$\phi \rightarrow \| \phi \| _ \rho ^ {\prime\prime} = \ \sup _ {\begin{array}{c} | y| < \rho , \\ x \in \mathbf R ^ {n} \end{array} } \ e ^ {\rho | x| } | \phi ( x + i y ) | ;$$

the imbedding $\Phi _ \rho \subset \Phi _ {\rho ^ \prime }$, $\rho > \rho ^ \prime$, is compact. Let $\Phi$ be the inductive limit of the (increasing) sequence of spaces $\Phi _ {1/n}$, $n \rightarrow \infty$. The space $\Phi$ is of type DFS, and its dual $\Phi ^ \prime$ is of type FS. The elements of $\Phi$ are Fourier hyperfunctions; $\Phi ^ \prime$ is also isomorphic to the space $S _ {1} ^ {1}$.

How to Cite This Entry:
Generalized functions, space of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_functions,_space_of&oldid=52034
This article was adapted from an original article by V.S. Vladimirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article