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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A kernel theorem on the space $[H_ \mu \times A;B]$
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by E. L. Koh and C. K. Li PDF
Proc. Amer. Math. Soc. 123 (1995), 177-182 Request permission

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.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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