A linear and continuous operator between Banach spaces is said to be absolutely summing if it maps unconditionally convergent series into absolutely convergent series. Moreover, it improves properties of stochastic processes. Indeed, N.Ghoussoub in [7] proved that an operator is absolutely summing if and only if it maps amarts (asymptotic martingales) into uniform amarts. In this paper we go a ...

A convergence in Riesz spaces is given axiomatically. A Bochner-type integral for Riesz space-valued functions is introduced and some Vitali and Lebesgue dominated convergence theorems are proved. Some properties and examples are investigated. A.M.S. SUBJECT CLASSIFICATION (2000): 28B15.

C n1ðfÞð f1 q1Þ þ n2ðfÞð f2 q2Þ jf qj f ðfÞdHðfÞ; q R oX; has continuous limit values on C if the truncated integrals.

Some additive reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in Hilbert spaces are given. Applications for complex-valued functions are provided as well.

The continuity for some multilinear operators generated by certain integral operators and Lipschitz functions on some Hardy and Herz-type spaces are obtained. The operators include Littlewood-Paley operators, Marcinkiewicz operators and Bochner-Riesz operator.