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.