Nuclear bilinear form
A bilinear form on the Cartesian product
of two locally convex spaces
and
that can be represented as
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where is a summable sequence,
and
are equicontinuous sequences (cf. Equicontinuity) in the dual spaces
and
of
and
, respectively, and
denotes the value of the linear functional
on the vector
. All nuclear bilinear forms are continuous. If
is a nuclear space, then for any locally convex space
all continuous bilinear forms on
are nuclear (the kernel theorem). This result is due to A. Grothendieck [1]; the form stated is given in [2]; for other statements see [3]. The converse holds: If a space
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 [4]. Let 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
consists of all generalized functions on the line. In the special case when
, the kernel theorem is equivalent to the following assertion: Every continuous bilinear functional on
has the form
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where and
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 on
can be identified with a continuous linear operator
by using the equality
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and this leads to Schwartz' kernel theorem: For any continuous linear mapping there is a unique generalized function
such that
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for all . In other words,
is an integral operator with kernel
.
References
[1] | A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc. (1955) |
[2] | A. Pietsch, "Nuclear locally convex spaces" , Springer (1972) (Translated from German) |
[3] | I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1964) (Translated from Russian) |
[4] | L. Schwartz, "Théorie des noyaux" , Proc. Internat. Congress Mathematicians (Cambridge, 1950) , 1 , Amer. Math. Soc. (1952) pp. 220–230 |
[5] | L. Schwartz, "Espaces de fonctions différentielles à valeurs vectorielles" J. d'Anal. Math. , 4 (1954–1955) pp. 88–148 |
Comments
References
[a1] | F. Trèves, "Topological vectorspaces, distributions and kernels" , Acad. Press (1967) |
[a2] | L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1966) |
Nuclear bilinear form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nuclear_bilinear_form&oldid=48024