A simple natural approach to the uniform boundedness principle for mutilinear mappings

• Bernandino, A. Thiago ^[1]
  1. [1] Universidade Federal de Pernambuco

     Universidade Federal de Pernambuco

     Brasil

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 28, Nº. 3, 2009, págs. 203-207
• Idioma: inglés
• DOI: 10.4067/S0716-09172009000300001
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• Resumen

  • The goal of this note is to give a new, simple and elegant proof to the Uniform Boundedness Principle (UBP) to m-linear mappings, which surprisingly, as far as we know, does not appear in the literature. The multilinear UBP is well-known for specialists but the original proof (presented in [4]) seems a little bit unnatural and uses the linear UBP. In the present note we show a quite simple argument which does not need to invoke the linear UBP and, when m = 1, recovers the classical proof of the linear case. As an immediate consequence, we obtain the Banach-Steinhaus Theorem (BST) for multilinear mappings.

• Referencias bibliográficas
  • Citas [1] A. Defant and K. Floret, Tensor Norms and Operators Ideals, North-Holland Mathematics Studies, 176, North-Holland, (1993).
  • [2] C. Fernandez, The closed graph theorem for multilinear mappings, International Journal of Mathematics and Mathematical Sciences, 19, pp....
  • [3] J. Mujica, Complex Analysis in Banach spaces, North-Holland Mathematics Studies 120, North-Holland, (1986).
  • [4] I. Sandberg, Multilinear maps and uniform boundedness. IEEE Trans. Circuits and Systems 32, pp. 332—336, (1985).