Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Representations for transformations continuous in the $ {\rm BV}$ norm


Authors: J. R. Edwards and S. G. Wayment
Journal: Trans. Amer. Math. Soc. 154 (1971), 251-265
MSC: Primary 28.50; Secondary 46.00
DOI: https://doi.org/10.1090/S0002-9947-1971-0274704-4
MathSciNet review: 0274704
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Riemann and Lebesgue-type integrations can be employed to represent operators on normed function spaces whose norms are not stronger than sup-norm by $ T(f) = \smallint f\,d\mu $ where $ \mu $ is determined by the action of T on the simple functions. The real-valued absolutely continuous functions on [0, 1] are not in the closure of the simple functions in the BV norm, and hence such an integral representation of an operator is not obtainable. In this paper the authors develop a v-integral whose structure depends on fundamental functions different than simple functions. This integral is as computable as the Riemann integral. By using these fundamental functions, the authors are able to obtain a direct, analytic representation of the linear functionals on AC which are continuous in the BV norm in terms of the v-integral. Further, the v-integral gives a characterization of the dual of AC in terms of the space of fundamentally bounded set functions which are convex with respect to length. This space is isometrically isomorphically identified with the space of Lipschitz functions anchored at zero with the norm given by the Lipschitz constant, which in turn is isometrically isomorphic to $ {L^\infty }$. Hence a natural identification exists between the classical representation and the one given in this paper. The results are extended to the vector setting.


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

  • [1] N. Dunford and J. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [2] R. J. Easton and D. H. Tucker, A generalized Lebesgue-type integral, Math. Ann. 181 (1969), 311-324. MR 1513279
  • [3] J. R. Edwards and S. G. Wayment, A unifying integral representation theorem, Math. Ann. 187 (1970), 317-328. MR 0270181 (42:5074)
  • [4] -, Integral representations for continuous linear operators in the setting of convex topological vector spaces, Trans. Amer. Math. Soc. (to appear). MR 0281867 (43:7581)
  • [5] T. H. Hildebrandt, Linear continuous functionals on the space (BV) with weak topologies, Proc. Amer. Math. Soc. 17 (1966), 658-664. MR 33 #1710. MR 0193490 (33:1710)
  • [6] -, Linear operations on functions of bounded variation, Bull. Amer. Math. Soc. 44 (1938), 75. MR 1563684
  • [7] R. Lane, The integral of a function with respect to a function, Proc. Amer. Math. Soc. 5 (1954), 59-66. MR 15, 514. MR 0059346 (15:514e)
  • [8] I. P. Natanson, Theory of functions of a real variable, GITTL, Moscow, 1957; English transl., Ungar, New York, 1961. MR 26 #6309. MR 0039790 (12:598d)
  • [9] E. J. Purcell, Calculus with analytic geometry, Appleton-Century-Crofts, New York, 1965.
  • [10] F. Riesz, Sur les opérations fonctionelles linéaires, C. R. Acad. Sci. Paris 149 (1909), 974-977.
  • [11] D. H. Tucker, A note on the Riesz representation theorem, Proc. Amer. Math. Soc. 14 (1963), 354-358. MR 26 #2865. MR 0145334 (26:2865)
  • [12] D. H. Tucker and S. G. Wayment, Absolute continuity and the Radon-Nikodym theorem, J. Reine Angew. Math. (to appear). MR 0272978 (42:7859)
  • [13] D. H. Uherka, Generalized Stieltjes integrals and a strong representation theorem for continuous maps on a function space, Math. Ann. 182 (1969), 60-66. MR 0247439 (40:705)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 28.50, 46.00

Retrieve articles in all journals with MSC: 28.50, 46.00


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0274704-4
Keywords: Generalized Stieltjes-type integral, v-integral, integral representation, convex with respect to length, fundamentally bounded, Lipschitz function, dual space, vector-valued functions, absolute continuity, bounded variation, semiabsolute continuity, semibounded variation
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society