Consistency theorems for almost convergence
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- by G. Bennett and N. J. Kalton PDF
- Trans. Amer. Math. Soc. 198 (1974), 23-43 Request permission
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 Lorentzโs investigations and obtain consistency theorems for almost convergence; this is achieved by studying certain locally convex topological vector spaces.References
- A. Alexiewicz and Z. Semadeni, Linear functionals on two-norm spaces, Studia Math. 17 (1958), 121โ140. MR 100211, DOI 10.4064/sm-17-2-121-140 S. Banach, Thรฉorie des operations linรฉaires, Monografie Mat., PWN, Warsaw, 1932. S. Banach and S. Saks, Sur la convergence forte dans les champs ${L^p}$, Studia Math. 2 (1930), 51-57.
- G. Bennett, A representation theorem for summability domains, Proc. London Math. Soc. (3) 24 (1972), 193โ203. MR 291685, DOI 10.1112/plms/s3-24.2.193
- G. Bennett, A new class of sequence spaces with applications in summability theory, J. Reine Angew. Math. 266 (1974), 49โ75. MR 344846, DOI 10.1515/crll.1974.266.49
- G. Bennett and N. J. Kalton, $FK$-spaces containing $c_{0}$, Duke Math. J. 39 (1972), 561โ582. MR 310597 โ, Inclusion theorems for $K$-spaces, Canad. J. Math. (to appear).
- G. Bennett and N. J. Kalton, Addendum to: โ$FK$-spaces containing $c_{0}$โ, Duke Math. J. 39 (1972), 819โ821. MR 313758 M.M. Day, Normed linear spaces, 2nd rev. ed., Academic Press, New York; Springer-Verlag, Berlin, 1962. MR 26 #2847.
- D. J. H. Garling, On topological sequence spaces, Proc. Cambridge Philos. Soc. 63 (1967), 997โ1019. MR 218880, DOI 10.1017/s0305004100042031
- N. J. Kalton, Some forms of the closed graph theorem, Proc. Cambridge Philos. Soc. 70 (1971), 401โ408. MR 301476, DOI 10.1017/s0305004100050039
- J. P. King, Almost summable sequences, Proc. Amer. Math. Soc. 17 (1966), 1219โ1225. MR 201872, DOI 10.1090/S0002-9939-1966-0201872-6
- G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167โ190. MR 27868, DOI 10.1007/BF02393648
- S. Mazur and W. Orlicz, On linear methods of summability, Studia Math. 14 (1954), 129โ160 (1955). MR 68012, DOI 10.4064/sm-14-2-129-160
- Charles W. McArthur, On a theorem of Orlicz and Pettis, Pacific J. Math. 22 (1967), 297โ302. MR 213848
- Charles W. McArthur and James R. Retherford, Uniform and equicontinuous Schauder bases of subspaces, Canadian J. Math. 17 (1965), 207โ212. MR 173934, DOI 10.4153/CJM-1965-020-5
- G. M. Petersen, Almost convergence and the Buck-Pollard property, Proc. Amer. Math. Soc. 11 (1960), 469โ477. MR 111961, DOI 10.1090/S0002-9939-1960-0111961-7
- Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
- Paul Schaefer, Almost convergent and almost summable sequences, Proc. Amer. Math. Soc. 20 (1969), 51โ54. MR 235340, DOI 10.1090/S0002-9939-1969-0235340-5
- A. K. Snyder, Conull and coregular $\textrm {FK}$ spaces, Math. Z. 90 (1965), 376โ381. MR 185315, DOI 10.1007/BF01112357
- I. Tweddle, Vector-valued measures, Proc. London Math. Soc. (3) 20 (1970), 469โ489. MR 259065, DOI 10.1112/plms/s3-20.3.469
- Albert Wilansky, Topics in functional analysis, Lecture Notes in Mathematics, No. 45, Springer-Verlag, Berlin-New York, 1967. Notes by W. D. Laverell. MR 0223854
- A. Wiweger, Linear spaces with mixed topology, Studia Math. 20 (1961), 47โ68. MR 133664, DOI 10.4064/sm-20-1-47-68
- Karl Zeller, Allgemeine Eigenschaften von Limitierungsverfahren, Math. Z. 53 (1951), 463โ487 (German). MR 39824, DOI 10.1007/BF01175646
- Karl Zeller, Abschnittskonvergenz in $FK$-Rรคumen, Math. Z. 55 (1951), 55โ70 (German). MR 47799, DOI 10.1007/BF01212667
- K. Zeller and W. Beekmann, Theorie der Limitierungsverfahren, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 15, Springer-Verlag, Berlin-New York, 1970 (German). Zweite, erweiterte und verbesserte Auflage. MR 0264267
Additional Information
- © Copyright 1974 American Mathematical Society
- 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