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''distribution space''

The space dual to the space of test (sufficiently good) functions. The Fréchet–Schwartz spaces (cf. [[Fréchet space|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.

===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|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} $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Schwartz,   "Théorie des distributions" , '''1–2''' , Hermann  (1950–1951)  {{MR|2067351}} {{MR|0209834}} {{MR|0117544}} {{MR|0107812}} {{MR|0041345}} {{MR|0035918}} {{MR|0032815}} {{MR|0031106}} {{MR|0025615}} {{ZBL|0962.46025}} {{ZBL|0653.46037}} {{ZBL|0399.46028}} {{ZBL|0149.09501}} {{ZBL|0085.09703}} {{ZBL|0089.09801}} {{ZBL|0089.09601}} {{ZBL|0078.11003}} {{ZBL|0042.11405}} {{ZBL|0037.07301}} {{ZBL|0039.33201}} {{ZBL|0030.12601}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Topological vector spaces" , Addison-Wesley  (1977)  (Translated from French)  {{MR|0583191}} {{ZBL|1106.46003}} {{ZBL|1115.46002}} {{ZBL|0622.46001}} {{ZBL|0482.46001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Dieudonné,   L. Schwartz,   "La dualité dans les espaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384092.png" />) et (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384093.png" />)"  ''Ann. Inst. Fourier'' , '''1'''  (1949)  pp. 61–101  {{MR|38553}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Grothendieck,   "Sur les espaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384094.png" /> et <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384095.png" />"  ''Summa Brasil. Math.'' , '''3''' :  6  (1954)  pp. 57–123  {{MR|75542}} {{ZBL|0058.09803}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.M. Gel'fand,   G.E. Shilov,   "Generalized functions" , '''2''' , Acad. Press  (1968)  (Translated from Russian)  {{MR|435832}} {{ZBL|0159.18301}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Yoshinaga,   "On a locally convex space introduced by J.S.E. Silva"  ''J. Sci. Hiroshima Univ. Ser. A'' , '''21'''  (1957)  pp. 89–98  {{MR|0097702}} {{ZBL|0080.31303}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  T. Kawai,   "On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients"  ''J. Fac. Sci. Univ. Tokyo Sect. 1A Math.''  (1970)  pp. 467–517  {{MR|0298200}} {{ZBL|0212.46101}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.S. Vladimirov,   "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)  {{MR|0564116}} {{MR|0549767}} {{ZBL|0515.46034}} {{ZBL|0515.46033}} </TD></TR></table>

====Comments====
For functional-analytic properties of distribution spaces see [[#References|[a1]]].

For generalized function spaces which are invariant under certain given integral transformations see [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]].

Test spaces and spaces of generalized functions which satisfy such invariance requirements can be constructed starting from a separable [[Hilbert space|Hilbert space]] and an unbounded [[Self-adjoint operator|self-adjoint operator]]  $  A $
on  $  X $. 
The test space ( $  = $
analyticity space) is defined by  $  S _ {X,A} = \cup _ {t > 0 }  e  ^ {-} tA ( X) $. 
The distribution space  $  T _ {X,A} $ (= trajectory space) consists of mappings  $  F : ( 0 , \infty ) \rightarrow X $
with the property: for all  $  t , \tau > 0 $: 
$  F ( t + \tau ) = e  ^ {-} tA F ( \tau ) $. 
The duality pairing is  $  \langle  F , \phi \rangle = ( F ( \epsilon ) , e ^ {\epsilon A } \phi ) _ {X} $, 
where  $  \epsilon $
is sufficiently small and depends on  $  \phi \in S _ {X,A} $.

Both spaces are inductive and projective limits of Hilbert spaces. Many Gel'fand–Shilov spaces are of this type.

For classical examples, topological properties and operator algebras on those spaces see [[#References|[a3]]], [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Treves,   "Topological vectorspaces, distributions and kernels" , Acad. Press  (1967)  {{MR|225131}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.H. Zemanian,   "Generalized integral transformations" , Interscience  (1968)  {{MR|0423007}} {{ZBL|0181.12701}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.G. de Bruijn,   "A theory of generalized functions with applications to Wigner distribution and Weyl correspondence"  ''Niew Archief for Wiskunde (3)'' , '''21'''  (1973)  pp. 205–280  {{MR|}} {{ZBL|0269.46033}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.J.L. van Eijndhoven,   J. de Graaf,   "Trajectory spaces, generalized functions and unbounded operators" , ''Lect. notes in math.'' , '''1162''' , Springer  (1985)  {{MR|}} {{ZBL|0622.46032}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Antosik,   J. Mikusiński,   R. Sikorski,   "Theory of distributions. The sequential approach" , Elsevier  (1973)  {{MR|0365130}} {{ZBL|0267.46028}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Köthe,   "Topological vector spaces" , '''1''' , Springer  (1969)  {{MR|0248498}} {{MR|0178335}} {{ZBL|0179.17001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. Horvath,   "Topological vector spaces and distributions" , Addison-Wesley  (1966)  {{MR|0205028}} {{ZBL|0143.15101}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W. Rudin,   "Functional analysis" , McGraw-Hill  (1974)  {{MR|1157815}} {{MR|0458106}} {{MR|0365062}} {{ZBL|0867.46001}} {{ZBL|0253.46001}} </TD></TR></table>