Sequential-Evaluation Convergence Of Mapping Series

λ(X)-evaluation convergence of mapping series is the normal form for convergence of series in every area, and the study on its relationship or invariant is an important content of functional analysis. Based on the studies of multiplier convergent operator series, many mathematicians started to study evaluation convergence of operator series.Taking out the linear condition, normal evaluation convergence of mapping series has been studied recently.On these Basis, the paper mainly analyses the invariants of four kinds of sequential-evaluation convergent mapping series in abstract dual systems by takingλ(X) as l_p(X), c₀(X), c(X), l_∞(X). Moreover, the strongest instrinsic meaning ofλ(X)- evaluation convergence is presented, and important theorems of sequential ev- aluation convergent mapping series are improved in this paper. In addition, some invariants ofλ(X)-evaluation convergence in totally bounded subsets are discussed in this paper.Secondly, the applicable value of the strongest topology of l_p(X)( c₀(X), c(X) or l_∞(X))-evaluation convergent mapping series is discussed in abstract dual systems. Moreover, the largest invariant range ofλ(X)-evaluation convergence of mapping series is firstly presented in this paper.The invariant theory of series, Banach theorem, open mapping principle and uniform boundedness principle found the base of the locally convex space theory. The theories on sequential-evaluation convergent mapping series are core contents of functional analysis. They also offer a new breakthrough point for the material progress of functional analysis.