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

Authors

  • A. Thiago Bernandino Universidade Federal de Pernambuco.

DOI:

https://doi.org/10.4067/S0716-09172009000300001

Abstract

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.

Author Biography

A. Thiago Bernandino, Universidade Federal de Pernambuco.

Departamento de Matemática.

References

[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. 407-408, (1996).

[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).

How to Cite

[1]
A. T. Bernandino, “A simple natural approach to the uniform boundedness principle for mutilinear mappings”, Proyecciones (Antofagasta, On line), vol. 28, no. 3, pp. 203-207, 1.

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Section

Artículos