Vector-Valued Integration in $BK$-Spaces

Abstract

Questions of convergence in $B$K-spaces, i.e. Banach spaces of complex-valued sequences $x=(x_k{)}_{k\in \mathbb{Z}}$ with continuity of all functionals $x\mapsto x_k((k\in \mathbb{Z})$ will be studied by methods of Fourier analysis. An elegant treatment is possible if the Cesàro sections of a $BK$-space element $x$ can be represented by vector-valued Riemann integrals. This was done by Goes [2] following the example of Katznelson [5: pp. 10-12). The purpose of this paper is to make precise the conditions in [2) concerning Riemann integration and to demonstrate relations between $BK$-spaces which are generated by a given $BK$-space.