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Transactions of the American Mathematical Society

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Convergence of Polynomial Level Sets


Author: J. Ferrera
Journal: Trans. Amer. Math. Soc. 350 (1998), 4757-4773
MSC (1991): Primary 46E25; Secondary 12E05
DOI: https://doi.org/10.1090/S0002-9947-98-02342-3
MathSciNet review: 1615959
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Abstract: In this paper we give a characterization of pointwise and uniform convergence of sequences of homogeneous polynomials on a Banach space by means of the convergence of their level sets. Results are obtained both in the real and the complex cases, as well as some generalizations to the nonhomogeneous case and to holomorphic functions in the complex case. Kuratowski convergence of closed sets is used in order to characterize pointwise convergence. We require uniform convergence of the distance function to get uniform convergence of the sequence of polynomials.


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

  • [B-P] M. Baronti and P. Papini, Convergence of sequences of sets. In Methods of functional analysis in approximation theory, ISNM 76, Birkhäuser, Basel, 1986, pp. 133-155. MR 88i:46028
  • [B] G. Beer, Convergence of continuous linear functionals and their level sets, Arch. Math. 52 (1989), 482-491. MR 90i:46018
  • [B-V] J. M. Borwein and J. Vanderwerff, Dual Kadec-Klee norms and the relationships between Wijsman, slice and Mosco convergence, Michigan Math. J. 41 (1994), 371-387. MR 95e:46015
  • [D] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1984. MR 85i:46020
  • [K] C. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966. MR 36:840
  • [Ll] J. Llavona, Approximation of continously differentiable functions, North-Holland Math. Studies, vol. 130, North-Holland, Amsterdam, 1986. MR 88f:41001
  • [M] J. Mujica, Complex analysis in Banach spaces, North-Holland Math. Studies, vol. 120, North-Holland, Amsterdam, 1986. MR 88d:46084

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Additional Information

J. Ferrera
Affiliation: Departmento Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, España
Email: ferrera@eucmax.sim.ucm.es

DOI: https://doi.org/10.1090/S0002-9947-98-02342-3
Keywords: Polynomials in Banach spaces, set convergence, level sets
Received by editor(s): March 20, 1995
Additional Notes: Research partially supported by DGICYT grant PB-0044 (Spain).
Article copyright: © Copyright 1998 American Mathematical Society

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