Factorization of weakly continuous holomorphic mappings

Manuel González; Joaqín Gutiérrez

Studia Mathematica (1996)

  • Volume: 118, Issue: 2, page 117-133
  • ISSN: 0039-3223

Abstract

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We prove a basic property of continuous multilinear mappings between topological vector spaces, from which we derive an easy proof of the fact that a multilinear mapping (and a polynomial) between topological vector spaces is weakly continuous on weakly bounded sets if and only if it is weakly uniformly} continuous on weakly bounded sets. This result was obtained in 1983 by Aron, Hervés and Valdivia for polynomials between Banach spaces, and it also holds if the weak topology is replaced by a coarser one. However, we show that it need not be true for a stronger topology, thus answering a question raised by Aron. As an application of the first result, we prove that a holomorphic mapping ƒ between complex Banach spaces is weakly uniformly continuous on bounded subsets if and only if it admits a factorization of the form f = g∘S, where S is a compact operator and g a holomorphic mapping.

How to cite

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González, Manuel, and Gutiérrez, Joaqín. "Factorization of weakly continuous holomorphic mappings." Studia Mathematica 118.2 (1996): 117-133. <http://eudml.org/doc/216267>.

@article{González1996,
abstract = {We prove a basic property of continuous multilinear mappings between topological vector spaces, from which we derive an easy proof of the fact that a multilinear mapping (and a polynomial) between topological vector spaces is weakly continuous on weakly bounded sets if and only if it is weakly uniformly\} continuous on weakly bounded sets. This result was obtained in 1983 by Aron, Hervés and Valdivia for polynomials between Banach spaces, and it also holds if the weak topology is replaced by a coarser one. However, we show that it need not be true for a stronger topology, thus answering a question raised by Aron. As an application of the first result, we prove that a holomorphic mapping ƒ between complex Banach spaces is weakly uniformly continuous on bounded subsets if and only if it admits a factorization of the form f = g∘S, where S is a compact operator and g a holomorphic mapping.},
author = {González, Manuel, Gutiérrez, Joaqín},
journal = {Studia Mathematica},
keywords = {weakly continuous holomorphic mapping; factorization of holomorphic mappings; polynomial; weakly continuous multilinear mapping; continuous multilinear mappings; weakly continuous on weakly bounded sets; weakly uniformly continuous},
language = {eng},
number = {2},
pages = {117-133},
title = {Factorization of weakly continuous holomorphic mappings},
url = {http://eudml.org/doc/216267},
volume = {118},
year = {1996},
}

TY - JOUR
AU - González, Manuel
AU - Gutiérrez, Joaqín
TI - Factorization of weakly continuous holomorphic mappings
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 2
SP - 117
EP - 133
AB - We prove a basic property of continuous multilinear mappings between topological vector spaces, from which we derive an easy proof of the fact that a multilinear mapping (and a polynomial) between topological vector spaces is weakly continuous on weakly bounded sets if and only if it is weakly uniformly} continuous on weakly bounded sets. This result was obtained in 1983 by Aron, Hervés and Valdivia for polynomials between Banach spaces, and it also holds if the weak topology is replaced by a coarser one. However, we show that it need not be true for a stronger topology, thus answering a question raised by Aron. As an application of the first result, we prove that a holomorphic mapping ƒ between complex Banach spaces is weakly uniformly continuous on bounded subsets if and only if it admits a factorization of the form f = g∘S, where S is a compact operator and g a holomorphic mapping.
LA - eng
KW - weakly continuous holomorphic mapping; factorization of holomorphic mappings; polynomial; weakly continuous multilinear mapping; continuous multilinear mappings; weakly continuous on weakly bounded sets; weakly uniformly continuous
UR - http://eudml.org/doc/216267
ER -

References

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  15. [15] J. G. Llavona, Approximation of Continuously Differentiable Functions, Math. Stud. 130, North-Holland, Amsterdam, 1986. Zbl0642.41001
  16. [16] L. A. Moraes, Extension of holomorphic mappings from E to E'', Proc. Amer. Math. Soc. 118 (1993), 455-461. Zbl0796.46033
  17. [17] J. Mujica, Complex Analysis in Banach Spaces, Math. Stud. 120, North-Holland, Amsterdam, 1986. Zbl0586.46040
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