A kernel theorem on the space

Authors:
E. L. Koh and C. K. Li

Journal:
Proc. Amer. Math. Soc. **123** (1995), 177-182

MSC:
Primary 46E40; Secondary 46F12, 46M15

MathSciNet review:
1217454

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Abstract: Recently, we introduced a space which consists of Banach space-valued distributions for which the Hankel transformation is an automorphism (*The Hankel transformation of a Banach space-valued generalized function*, Proc. Amer. Math. Soc. **119** (1993), 153-163). One of the cornerstones in distribution theory is the kernel theorem of Schwartz which characterizes continuous bilinear functionals as kernel operators. The object of this paper is to prove a kernel theorem which states that for an arbitrary element of , it can be uniquely represented by an element of and hence of . This is motivated by a generalization of Zemanian (*Realizability theory for continuous linear systems*, Academic Press, New York, 1972) for the product space where *V* is a Fréchet space. His work is based on the facts that the space is an inductive limit space and the convolution product is well defined in . This is not possible here since the space is not an inductive limit space. Furthermore, is not dense in . To overcome this, it is necessary to apply some results from our aforementioned paper. We close this paper with some applications to integral transformations by a suitable choice of *A*.

**[1]**A. H. Zemanian,*Generalized integral transformations*, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1968. Pure and Applied Mathematics, Vol. XVIII. MR**0423007****[2]**A. H. Zemanian,*Realizability theory for continuous linear systems*, Academic Press, New York-London, 1972. Mathematics in Science and Engineering, Vol. 97. MR**0449807****[3]**E. L. Koh and C. K. Li,*The Hankel transformation of Banach-space-valued generalized functions*, Proc. Amer. Math. Soc.**119**(1993), no. 1, 153–163. MR**1149972**, 10.1090/S0002-9939-1993-1149972-7

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DOI:
https://doi.org/10.1090/S0002-9939-1995-1217454-1

Keywords:
Kernel Theorem,
the Hankel transformation,
inductive-limit topology,
generalized function

Article copyright:
© Copyright 1995
American Mathematical Society