View source for Generalized functions, space of

''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g0438401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g0438402.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g0438404.png" /> of (rapidly-decreasing) test functions consists of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g0438405.png" />-functions that together with all their derivatives decrease at infinity faster than any power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g0438406.png" />. This space is the projective limit of the sequence of Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g0438407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g0438408.png" /> consisting of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g0438409.png" />-functions with norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384010.png" /></td> </tr></table>

and the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384011.png" /> is compact; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384012.png" /> is of type FS. The dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384014.png" /> (the space of generalized functions of slow growth) is the inductive limit of the sequence of Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384015.png" />, where the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384016.png" /> is compact, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384017.png" /> is of type DFS. If a sequence of generalized functions is (weakly) convergent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384018.png" />, then it converges with respect to the norm of functionals in some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384019.png" />. The Fourier transformation is an isomorphism on the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384021.png" />.

2) The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384026.png" /> an open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384027.png" />). The space of test functions consists of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384028.png" />-functions that have compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384029.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384030.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384031.png" /> of type FS, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384032.png" /> is a strictly-increasing sequence of open sets that exhausts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384035.png" /> compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384036.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384037.png" /> is the projective limit of the (decreasing) sequence of Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384039.png" /> consisting of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384040.png" /> functions with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384041.png" /> and with norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384042.png" /></td> </tr></table>

where the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384043.png" /> is compact. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384044.png" /> be the space (strongly) dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384045.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384047.png" />. A sequence of test functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384048.png" /> converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384049.png" /> if it converges in some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384050.png" />. A sequence of generalized functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384051.png" /> converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384052.png" /> if it converges on every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384053.png" /> (weak convergence). For a linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384054.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384055.png" /> to be a generalized function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384056.png" /> it is necessary and sufficient that for any open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384057.png" /> there exist numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384059.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384060.png" /></td> </tr></table>

The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384061.png" /> is (weakly) complete: If a sequence of generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384063.png" /> is such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384064.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384065.png" /> the sequence of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384066.png" /> converges, then the functional

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384067.png" /></td> </tr></table>

belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384068.png" />. A generalized function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384069.png" /> has unrestricted  "growth"  in a neighbourhood of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384070.png" />; in particular, any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384071.png" /> determines a generalized function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384072.png" /> by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384073.png" /></td> </tr></table>

3) The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384075.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384076.png" /> be the Banach space of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384078.png" />, that are holomorphic in the tubular neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384080.png" />, with norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384081.png" /></td> </tr></table>

the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384083.png" />, is compact. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384084.png" /> be the inductive limit of the (increasing) sequence of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384086.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384087.png" /> is of type DFS, and its dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384088.png" /> is of type FS. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384089.png" /> are Fourier hyperfunctions; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384090.png" /> is also isomorphic to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384091.png" />.

====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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384096.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384097.png" />. The test space (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384098.png" /> analyticity space) is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g04384099.png" />. The distribution space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g043840100.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g043840101.png" /> trajectory space) consists of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g043840102.png" /> with the property: for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g043840103.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g043840104.png" />. The duality pairing is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g043840105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g043840106.png" /> is sufficiently small and depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043840/g043840107.png" />.

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>