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


  • A. Thiago Bernandino Universidade Federal de Pernambuco.




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.


[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

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.