局部凸空间中的绝对收敛级数研究

A study on absolutely convergent series in locally convex spaces

在局部凸空间中建立了级数绝对收敛的概念,使得原本在赋范线性空间中的简单定义得到推广.在局部凸空间中得到了绝对收敛级数的等价性质,利用该局部凸空间的对偶空间中的等度连续集来完成对绝对收敛级数的等价描述.更进一步得到结论:利用从该局部凸空间到另一个局部凸空间的连续线性算子空间中的等度连续集也可以完成对绝对收敛级数的等价刻画.同时,在局部凸的准范空间中研究了绝对收敛级数,得到了级数的绝对收敛与空间上的准范数的一种内在联系.

The concept of absolute convergence for series, a previously simple concept in normed spaces, was generalized to locally convex spaces. In addition, the equivalent characterization for an absolutely convergent series in locally convex spaces was obtained, in which an equicontinuous set in the duality of the locally convex spaces was employed to characterize the absolutely convergent series. Then this result was generalized to an equicontinuous set in the space of all continuous linear operators from the locally convex space to another locally convex space. Meanwhile, absolutely convergent series were researched in paranormed locally convex spaces and an interrelationship between absolutely convergent series and the paranorm in the space was established .