On axioms of convergence in linear spaces
Abstract
By a (general) convergence in a given linear space X we mean a mapping G: XN→2X, where N denotes the set of all positive integers, and by a zero-convergence in X we mean a convergence G0 in X for which G0(x)≠⌀ implies 0∈G0(x) for each x∈XN. In the paper, the two operations are defined: 1° operation C, which to each zero-convergence G0 in X assigns some general convergence G in X, and 2° operation C0, which to each general convergence G in X assigns a zero-convergence G0 in X. Various systems of axioms for general convergences and zero-convergences are considered and their connections with the operations C and C0 are studied. Also mutual independence of axioms is studied.
References
2. J. Burzyk, On convergence in the Mikusiński operational calculus, Studia Math. (to appear).
3. J. Burzyk, On type II convergence in the Mikusiński operational calculus, Studia Math. (to appear).
4. A. Kamiński, On characterization of topological convergences, Proc. of the Conf. on Convergence - Szczyrk 1979, Katowice, 1980, 50-70.
5. A. Kamiński, On multivalued topological convergences, Bull. Acad. Polon. Sci. Sér. Sci. Math. 39 (1981), 607-610.
6. A. Kamiński, Remarks on multivalued convergences, Proc. of the 5th Prague Topological Symposium - Prague 1981 (to appear).
7. L.V. Kantorovitch, B.Z. Vulih, A.G. Pinsker, Funkcionalnyi analiz v poluuporiadočennyh prostranstvah, Moskva 1950 (in Russian).
8. J. Kisyński, Convergence du type L, Colloq. Math. 12 (1960), 205-211.
9. J. Mikusiński, Operational Calculus, Pergamon Press - PWN, Warszawa 1959.
10. P. Mikusiński, Aksjomatyczna teoria zbieżności, [w:] Prace matematyczne 12. Prace naukowe Uniwersytetu Śląskiego nr 425, pod red. K. Szymiczka, Katowice 1982.
11. J. Novák, On convergence groups, Czechoslovak Math. J. 20 (1970), 357-374.
Instytut Matematyki, Polska Akademia Nauk Poland
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