Vector-valued integration in \(BK\)-spaces (Q1909639)

Summary: Questions of convergence in \(BK\)-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 \textit{G. Goes} [Stud. Math. 57, 241-249 (1976; Zbl 0346.42004)] following the example of \textit{Y. Katznelson} [``An introduction to harmonic analysis'' (1968; Zbl 0169.17902), pp. 10-12]. The purpose of this paper is to make precise the conditions in [\textit{Goes}, loc. cit.] concerning Riemann integration and to demonstrate relations between \(BK\)-spaces which are generated by a given \(BK\)-space.