This paper is a continuation of an earlier one [Part I, Acta Math. Hung. 55, 149-160 (1990; Zbl 0781.42006)] wherein the authors studied various modes of convergence of the Fourier partial sums \(\{s_ n f\}^ \infty_{n=0}\) of a continuous \(2\pi\)-periodic function and showed that the natural spaces generated by these modes are Banach spaces under several equivalent norms. In this paper, similar results are obtained for the space \(A^ \lambda\) [resp. \(S^ \lambda\)]\(=\{f\in L^ 1: \{s_ n f\}\) is absolutely convergent [resp. strongly convergent] of index \(\lambda\) a.e. to \(f\)\} for \(1\leq \lambda\).