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Consistency theorems for almost convergence


Authors: G. Bennett and N. J. Kalton
Journal: Trans. Amer. Math. Soc. 198 (1974), 23-43
MSC: Primary 46A45
DOI: https://doi.org/10.1090/S0002-9947-1974-0352932-X
MathSciNet review: 0352932
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Abstract: The concept of almost convergence of a sequence of real or complex numbers was introduced by Lorentz, who developed a very elegant theory. The purpose of the present paper is to continue Lorentz's investigations and obtain consistency theorems for almost convergence; this is achieved by studying certain locally convex topological vector spaces.


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  • [1] A. Alexiewicz and Z. Semadeni, Linear functionals on two-norm spaces, Studia Math. 17 (1958), 121-140. MR 20 #6644. MR 0100211 (20:6644)
  • [2] S. Banach, Théorie des operations linéaires, Monografie Mat., PWN, Warsaw, 1932.
  • [3] S. Banach and S. Saks, Sur la convergence forte dans les champs $ {L^p}$, Studia Math. 2 (1930), 51-57.
  • [4] G. Bennett, A representation theorem for summability domains, Proc. Lond. Math. Soc. (3) 24 (1972), 193-203. MR 45 #776. MR 0291685 (45:776)
  • [5] -, A new class of sequence spaces with applications in summability theory, J. Reine Angew. Math. 266 (1974), 49-75. MR 0344846 (49:9585)
  • [6] G. Bennett and N.J. Kalton, FK-spaces containing $ {c_0}$, Duke Math. J. 39 (1972), 561-582. MR 0310597 (46:9695)
  • [7] -, Inclusion theorems for $ K$-spaces, Canad. J. Math. (to appear).
  • [8] -, Addendum to FK-spaces containing $ {c_0}$, Duke Math. J. 39 (1972), 819-821. MR 0313758 (47:2312)
  • [9] M.M. Day, Normed linear spaces, 2nd rev. ed., Academic Press, New York; Springer-Verlag, Berlin, 1962. MR 26 #2847.
  • [10] D.J.H. Garling, On topological sequence spaces, Proc. Cambridge Philos. Soc. 63 (1967), 997-1019. MR 36 #1964. MR 0218880 (36:1964)
  • [11] N.J. Kalton, Some forms of the closed graph theorem, Proc. Cambridge Philos. Soc. 70 (1971), 401-408. MR 0301476 (46:634)
  • [12] J.P. King, Almost summable sequences, Proc. Amer. Math. Soc. 17 (1966), 1219-1225. MR 34 #1752. MR 0201872 (34:1752)
  • [13] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167-190. MR 10, 367. MR 0027868 (10:367e)
  • [14] S. Mazur and W. Orlicz, On linear methods of summability, Studia Math. 14 (1954), 129-160. MR 16, 814. MR 0068012 (16:814a)
  • [15] C.W. McArthur, On a theorem of Orlicz and Pettis, Pacific J. Math. 22 (1967), 297-302. MR 35 #4702. MR 0213848 (35:4702)
  • [16] C.W. McArthur and J.R. Retherford, Uniform and equicontinuous Schauder bases of subspaces, Canad. J. Math. 17 (1965), 207-212. MR 30 #4141. MR 0173934 (30:4141)
  • [17] G.M. Petersen, Almost convergence and the Buck-Pollard property, Proc. Amer. Math. Soc. 11 (1960), 469-477. MR 22 #2819. MR 0111961 (22:2819)
  • [18] H.H. Schaefer, Topological vector spaces, Macmillan, New York, 1966. MR 33 #1689. MR 0193469 (33:1689)
  • [19] P. Schaefer, Almost convergent and almost summable sequences, Proc. Amer. Math. Soc. 20 (1969), 51-54. MR 38 #3649. MR 0235340 (38:3649)
  • [20] A.K. Snyder, Conull and coregular FK-spaces, Math. Z. 90 (1965), 376-381. MR 32 #2783. MR 0185315 (32:2783)
  • [21] I. Tweddle, Vector-valued measures, Proc. London Math. Soc. (3) 20 (1970), 469-489. MR 41 #3707. MR 0259065 (41:3707)
  • [22] A. Wilansky, Topics in functional analysis, Lecture Notes in Math., no. 45, Springer-Verlag, Berlin and New York, 1967. MR 36 #6901. MR 0223854 (36:6901)
  • [23] A. Wiweger, Linear spaces with mixed topology, Studia Math. 20 (1961), 47-68. MR 24 #A3490. MR 0133664 (24:A3490)
  • [24] K. Zeller, Allgemeine Eigenschaften von Limitierungsverfahren, Math. Z. 53 (1951), 463-487. MR 12, 604. MR 0039824 (12:604e)
  • [25] -, Abschnittskonvergenz in FK-Räumen, Math. Z. 55 (1951), 55-70. MR 13, 934. MR 0047799 (13:934b)
  • [26] K. Zeller and W. Beekman, Theorie der Limitierungsverfahren. Zweite, erweiterte und verbesserte Auflage, Ergebnisse der Math. und ihrer Grenzgebiete, Band 15, Springer-Verlag, Berlin and New York, 1970. MR 41 #8863. MR 0264267 (41:8863)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0352932-X
Keywords: Banach limit, almost convergence, topological sequence space, FK-space, matrix transformation, consistency theorems, superconvergence in topological vector spaces
Article copyright: © Copyright 1974 American Mathematical Society

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