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A kernel theorem on the space $ [H\sb \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
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.

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Keywords: Kernel Theorem, the Hankel transformation, inductive-limit topology, generalized function
Article copyright: © Copyright 1995 American Mathematical Society