A kernel theorem on the space $[H_ \mu \times A;B]$

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

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

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

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

MathSciNet review:
1217454

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Abstract: Recently, we introduced a space $[{H_\mu }(A);B]$ 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 $[{H_\mu } \times A;B]$, it can be uniquely represented by an element of $[{H_\mu }(A);B]$ and hence of $[{H_\mu };[A;B]]$. This is motivated by a generalization of Zemanian (*Realizability theory for continuous linear systems*, Academic Press, New York, 1972) for the product space ${D_{{R^n}}} \times V$ where *V* is a Fréchet space. His work is based on the facts that the space ${D_{{R^n}}}$ is an inductive limit space and the convolution product is well defined in ${D_{{K_j}}}$. This is not possible here since the space ${H_\mu }(A)$ is not an inductive limit space. Furthermore, $D(A)$ is not dense in ${H_\mu }(A)$. 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*.

- 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** - A. H. Zemanian,
*Realizability theory for continuous linear systems*, Academic Press, New York-London, 1972. Mathematics in Science and Engineering, Vol. 97. MR**0449807** - 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**, DOI https://doi.org/10.1090/S0002-9939-1993-1149972-7

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

Article copyright:
© Copyright 1995
American Mathematical Society