Abstract
We prove a unified convergence theorem, which presents, in four equivalent forms, the famous Antosik–Mikusinski theorems. In particular, we show that Swartz’ three uniform convergence principles are all equivalent to the Antosik–Mikusinski theorems.
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Mikusinski, J., Sikorski, R.: The elementary theory of distributions I. Rozprawy Mat., 12, 123–125 (1957)
Mikusinski, J., Sikorski, R.: The elementary theory of distributions II. Rozprawy Mat., 25, 171–174 (1961)
Schwartz, L.: Theorie des distributions, Hermann, Paris, 1966
Mikusinski, J.: A theorem on vector matrices and its applications in measure theory and functional analysis. Bull. Acad. Polon. Sci., 18, 193–196 (1970)
Swartz, C.: Infinite matrices and the gliding hump, World Sci. Publ., Singapore, 1996
Antosik, P., Swartz, C.: Matrix methods in analysis, Springer Lecture Notes in Math., 1113, Heidelberg, 1985
Stuart, C.: Weak sequentially completeness in sequence spaces, Ph. D. Dissertation, New Mexico State University, 1993
Stuart, C.: Interchanging the limits in a double series. Southeast Asian Bull. Math., 18, 81–84 (1994)
Wu, J. D., Lu, S. J.: A summation theorem and its applications. J. Math. Anal. Appl., 257, 29–38 (2001)
Wu, J. D., Lu, S. J.: A full invariant theorem and some applications. J. Math. Anal. Appl., 270, 397–404 (2002)
Weber, H.: A diagonal theorem: Answer to a question of Antosik. Bull. Polish Acad. Sci. Math., 41, 95–102 (1993)
Li, R. L., Cho, M. Y.: A uniform convergent principle. J. Harbin Institute of Technology, 24, 107–108 (1992)
Li, R. L., Li, L. S., Kang, S. M.: Summability results for operator matrices on topological vector spaces. Sci. in China, Ser. A, 44, 1300–1311 (2001)
Qu, W. B., Wu, J. D.: On Antosik’s lemma and the Antosik–Mikusinski basic matrix theorem. Proc. Amer. Math. Soc., 130, 3283–3285 (2002)
Antosik, P.: A lemma on matrices and its applications. Contemporary Math., 52, 89–95 (1986)
Wu, J. D., Li, R. L., Qu, W. B.: The extension of Eberlein–Smulian theorem in locally convex spaces. Acta Math. Sinica, Chinese Series, 41, 663–666 (1998)
Thomas, G. E. F.: L’Integration par rapport a une mesure de Radon vectorielle. Ann. Inst. Fourier (Grenoble), 20, 55–191 (1970)
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This project is supported by NSFC (10471124) and is supported by Zhejiang Provincial Natural Science Foundation of China (M103057) and sponsored by SRF for ROCS, SEM
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Wu, J.D., Luo, J.W. & Lu, S.J. A Unified Convergence Theorem. Acta Math Sinica 21, 315–322 (2005). https://doi.org/10.1007/s10114-004-0481-5
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DOI: https://doi.org/10.1007/s10114-004-0481-5