CHARACTERIZATIONS OF ALMOST CONVERGENT SEQUENCES IN A HILBERT SPACE OR IN $L^p(T)$

Abstract

In [Acta Math. 80(1948),167-190], G. G. Lorentz characterized almost convergent sequences in $\mathbb R$ (or in $\mathbb C$) in terms of the concept of uniform convergence of the de la Vallée-Poussin means. In this paper, we give a further study on such kind of convergence for any Hilbert space or $L^p(T)$, where $1 \le p \le \infty$. Two new Cauchy forms for almost convergence are established. We prove that any of them is equivalent to the one established by Miller and Orhan. We use these forms to characterize almost convergent sequences in the aforementioned spaces in terms of coefficients.