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)

 
 

 

Steinhaus type theorems for summability matrices


Author: I. J. Maddox
Journal: Proc. Amer. Math. Soc. 45 (1974), 209-213
MSC: Primary 40C05; Secondary 40G10
DOI: https://doi.org/10.1090/S0002-9939-1974-0364938-0
MathSciNet review: 0364938
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Necessary and sufficient conditions are given for an infinite matrix to sum all bounded strongly summable sequences. It is shown that the Borel matrix does not sum all such sequences. A corollary is that the bounded summability field of the Borel method is strictly contained in that of the $ (C,1)$ method. Also, it is proved that no coregular matrix can almost sum all bounded sequences--a generalization of Steinhaus' theorem.


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

  • [1] D. Borwein, Linear functionals connected with strong Cesàro summability, J. London Math. Soc. 40 (1965), 628-634. MR 32 #2893. MR 0185426 (32:2893)
  • [2] G. H. Hardy, Divergent series, Clarendon Press, Oxford, 1949. MR 11, 25. MR 0030620 (11:25a)
  • [3] J. P. King, Almost summable sequences, Proc. Amer. Math. Soc. 17 (1966), 1219-1225. MR 34 #1752. MR 0201872 (34:1752)
  • [4] B. Kuttner, Note on strong summability, J. London Math. Soc. 21 (1946), 118-122. MR 8, 375. MR 0019141 (8:375c)
  • [5] G. G. Lorentz, A contribution to the theory of divergent series, Acta Math. 80 (1948), 167-190. MR 10, 367. MR 0027868 (10:367e)
  • [6] I. J. Maddox, On Kuttner's theorem, J. London Math. Soc. 43 (1968), 285-290. MR 37 #641. MR 0225044 (37:641)
  • [7] -, Elements of functional analysis, Cambridge Univ. Press, New York, 1970. MR 0390692 (52:11515)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 40C05, 40G10

Retrieve articles in all journals with MSC: 40C05, 40G10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0364938-0
Keywords: Bounded strongly summable sequences, infinite matrix, conservative matrix, Cesàro means, Borel matrix, summability, almost summability
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society