International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 3, Pages 285-296
A representation theorem for operators on a space of
Department of Mathematics, Southwest Texas State University, San Marcos 78666, Texas, USA
Received 4 May 1978
Copyright © 1978 J. A. Chatfield. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Suppose is a Banach space of norm and is the set of real numbers. All integrals used are of the subdivision-refinement type. The main theorem [Theorem 3] gives a representation of where is a function from to such that , , , and each exist for each and is a bounded linear operator on the space of all such functions . In particular we show that where each of , , and depend only on , is of bounded variation, and are except at a countable number of points, is a function from to depending on and denotes the points in . for which or . We also define an interior interval function integral and give a relationship between it and the standard interval function integral.