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1996 | 118 | 2 | 117-133
Tytuł artykułu

Factorization of weakly continuous holomorphic mappings

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EN
Abstrakty
EN
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.
Twórcy
  • Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, 39071 Santander, Spain, gonzalem@ccaix3.unican.es
  • Departamento de Matemática Aplicada, ETS de Ingenieros Industriales, Universidad Politécnica de Madrid, C. José Gutiérrez Abascal 2, 28006 Madrid, Spain, c0550003@ccupm.upm.es
Bibliografia
  • [1] R. M. Aron, Y. S. Choi and J. G. Llavona, Estimates by polynomials, Bull. Austral. Math. Soc. 52 (1995), 475-486.
  • [2] R. M. Aron, J. Gómez and J. G. Llavona, Homomorphisms between algebras of differentiable functions in infinite dimensions, Michigan Math. J. 35 (1988), 163-178.
  • [3] R. M. Aron, C. Hervés and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983), 189-204.
  • [4] R. M. Aron and J. B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195-216.
  • [5] R. M. Aron and M. Schottenloher, Compact holomorphic mappings on Banach spaces and the approximation property, J. Funct. Anal. 21 (1976), 7-30.
  • [6] A. Braunsz and H. Junek, Bilinear mappings and operator ideals, Rend. Circ. Mat. Palermo Suppl. (2) 10 (1985), 25-35.
  • [7] S. Dineen, Complex Analysis in Locally Convex Spaces, Math. Stud. 57, North-Holland, Amsterdam, 1981.
  • [8] S. Dineen, Entire functions on $c_0$, J. Funct. Anal. 52 (1983), 205-218.
  • [9] S. Dineen, Infinite Dimensional Complex Analysis, book in preparation.
  • [10] M. González and J. M. Gutiérrez, The compact weak topology on a Banach space, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 367-379.
  • [11] M. González and J. M. Gutiérrez, Weakly continuous mappings on Banach spaces with the Dunford-Pettis property, J. Math. Anal. Appl. 173 (1993), 470-482.
  • [12] S. Heinrich, Closed operator ideals and interpolation, J. Funct. Anal. 35 (1980), 397-411.
  • [13] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981.
  • [14] J. L. Kelley, General Topology, Grad. Texts in Math. 27, Springer, Berlin, 1955.
  • [15] J. G. Llavona, Approximation of Continuously Differentiable Functions, Math. Stud. 130, North-Holland, Amsterdam, 1986.
  • [16] L. A. Moraes, Extension of holomorphic mappings from E to E'', Proc. Amer. Math. Soc. 118 (1993), 455-461.
  • [17] J. Mujica, Complex Analysis in Banach Spaces, Math. Stud. 120, North-Holland, Amsterdam, 1986.
  • [18] H. Porta, Compactly determined locally convex topologies, Math. Ann. 196 (1972), 91-100.
  • [19] R. A. Ryan, Weakly compact holomorphic mappings on Banach spaces, Pacific J. Math. 131 (1988), 179-190.
  • [20] J. H. Webb, Sequential convergence in locally convex spaces, Proc. Cambridge Philos. Soc. 64 (1968), 341-364.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv118i2p117bwm
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