Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
MathSciNet review: 1217454
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E40, 46F12, 46M15

Retrieve articles in all journals with MSC: 46E40, 46F12, 46M15

Additional Information

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