View source for Nuclear bilinear form

A [[Bilinear form|bilinear form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n0678201.png" /> on the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n0678202.png" /> of two locally convex spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n0678203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n0678204.png" /> that can be represented as

<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/n/n067/n067820/n0678205.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n0678206.png" /> is a summable sequence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n0678207.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n0678208.png" /> are equicontinuous sequences (cf. [[Equicontinuity|Equicontinuity]]) in the dual spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n0678209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782012.png" />, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782013.png" /> denotes the value of the linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782014.png" /> on the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782015.png" />. All nuclear bilinear forms are continuous. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782016.png" /> is a [[Nuclear space|nuclear space]], then for any locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782017.png" /> all continuous bilinear forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782018.png" /> are nuclear (the kernel theorem). This result is due to A. Grothendieck [[#References|[1]]]; the form stated is given in [[#References|[2]]]; for other statements see [[#References|[3]]]. The converse holds: If a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782019.png" /> satisfies the kernel theorem, then it is a nuclear space.

For spaces of smooth functions of compact support, the kernel theorem was first obtained by L. Schwartz [[#References|[4]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782020.png" /> be the nuclear space of all infinitely-differentiable functions with compact support on the real line, equipped with the standard locally convex topology of Schwartz, so that the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782021.png" /> consists of all generalized functions on the line. In the special case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782022.png" />, the kernel theorem is equivalent to the following assertion: Every continuous bilinear functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782023.png" /> has the form

<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/n/n067/n067820/n06782024.png" /></td> </tr></table>

<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/n/n067/n067820/n06782025.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782027.png" /> is a generalized function in two variables. There are similar statements of the kernel theorem for spaces of smooth functions in several variables with compact support, for spaces of rapidly-decreasing functions, and for other specific nuclear spaces. Similar results are valid for multilinear forms.

A continuous bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782028.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782029.png" /> can be identified with a continuous linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782030.png" /> by using the equality

<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/n/n067/n067820/n06782031.png" /></td> </tr></table>

and this leads to Schwartz' kernel theorem: For any continuous linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782032.png" /> there is a unique generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782033.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/n/n067/n067820/n06782034.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782035.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782036.png" /> is an integral operator with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067820/n06782037.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc.  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Pietsch,   "Nuclear locally convex spaces" , Springer  (1972)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.M. Gel'fand,   N.Ya. Vilenkin,   "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Schwartz,   "Théorie des noyaux" , ''Proc. Internat. Congress Mathematicians (Cambridge, 1950)'' , '''1''' , Amer. Math. Soc.  (1952)  pp. 220–230</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L. Schwartz,   "Espaces de fonctions différentielles à valeurs vectorielles"  ''J. d'Anal. Math.'' , '''4'''  (1954–1955)  pp. 88–148</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Trèves,   "Topological vectorspaces, distributions and kernels" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Schwartz,   "Théorie des distributions" , '''1–2''' , Hermann  (1966)</TD></TR></table>