### Impacto

Ferrera Cuesta, Juan
(1998)
*Convergence of polynomial level sets.*
Transactions of the American Mathematical Society, 350
(12).
4757-4773..
ISSN 0002-9947

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Official URL: http://www.ams.org/journals/tran/1998-350-12/S0002-9947-98-02342-3/S0002-9947-98-02342-3.pdf

## 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.

Item Type: | Article |
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Uncontrolled Keywords: | Polynomials in Banach spaces; Set convergence; Level sets; Sequences of homogeneous polynomials on a Banach space |

Subjects: | Sciences > Mathematics > Mathematical analysis |

ID Code: | 15271 |

References: | 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. G. Beer, Convergence of continuous linear functionals and their level sets, Arch. Math. 52 (1989), 482–491. 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. J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1984. C. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966. J. Llavona, Approximation of continously differentiable functions, North-Holland Math. Studies, vol. 130, North-Holland, Amsterdam, 1986. J. Mujica, Complex analysis in Banach spaces, North-Holland Math. Studies, vol. 120, North-Holland, Amsterdam, 1986. |

Deposited On: | 18 May 2012 08:56 |

Last Modified: | 06 Feb 2014 10:20 |

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