On the Banach Principle

Radu Zaharopol

Abstract: We extend the Banach principle to sequences of operators which have as range an Archimedean Riesz space (the Riesz space does not have to be a space of classes of equivalence of measurable functions). The class of Riesz spaces for which our extension works is quite large. The role played in the classical Banach principle by the almost everywhere convergence is taken by the notion of individual convergence which we have introduced in an earlier work. The absence of a measure is compensated by the use of the $\sigma$-order continuous dual of the Dedekind completion of the Riesz space involved in the extension. In order to prove our extension, we obtain a characterization of individually convergent sequences which resembles a Cauchy condition.

Keywords: Archimedean Riesz spaces; Dedekind completion; $\sigma$-order continuous dual.

Classification (MSC2000): 47A35, 47A05, 46A40

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