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)

 
 

 

Continuous functions on countable compact ordered sets as sums of their increments


Author: Gadi Moran
Journal: Trans. Amer. Math. Soc. 256 (1979), 99-112
MSC: Primary 46E15; Secondary 40A30, 54C05
DOI: https://doi.org/10.1090/S0002-9947-1979-0546909-3
MathSciNet review: 546909
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Every continuous function from a countable compact linearly ordered set A into a Banach space V (vanishing at the least element of A ) admits a representation as a sum of a series of its increments (in the topology of uniform convergence). This series converges to no other sum under rearrangements of its terms. A uniqueness result to the problem of representation of a regulated real function on the unit interval as a sum of a continuous and a steplike function is derived.


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

  • [E] H. B. Enderton, Elements of set theory, Academic Press, New York, 1977. MR 0439636 (55:12522)
  • [F] L. Fuchs, Infinite abelian groups, Academic Press, New York, 1970. MR 0255673 (41:333)
  • [HK] A. Haar and D. König, Über einfach geordnete Mengen, J. Reine Angew. Math. 139 (1911), 16-28.
  • [Ha] H. Hadwiger, Über die Konvergenzarten unendlicher Reihen im Hilbertschen Raum, Math. Z. 47 (1941), 325-329. MR 0006373 (3:295a)
  • [GMW] C. Goffman, G. Moran and D. Waterman, The structure of regulated functions, Proc. Amer. Math. Soc. 57 (1976), 61-65. MR 0401993 (53:5816)
  • [Ke] J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1964. MR 0070144 (16:1136c)
  • [Ku] K. Kuratowski, Topology. I, II, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [McA] C. W. McArthur, On relationships amongst certain spaces of sequences in an arbitrary Banach space, Canad. J. Math. 8 (1956), 192-197. MR 0078662 (17:1227d)
  • [M] G. Moran, Of regulated and steplike functions, Trans. Amer. Math. Soc. 231 (1977), 249-257. MR 0499028 (58:17004)
  • [M1] -, On scattered compact ordered sets, Proc. Amer. Math. Soc. 75 (1979), 355-360. MR 532165 (81j:54051)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46E15, 40A30, 54C05

Retrieve articles in all journals with MSC: 46E15, 40A30, 54C05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0546909-3
Keywords: Countable compact ordered set, continuous function, increments, regulated function
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society