Separability of the space of continuous functions that are continuous in Λ-variation

The Shao–Sablin index of a Λ-sequence Λ = (λi ) is defined by SΛ := lim supn→∞ 2n i=1 1 λi / n i=1 1 λi . The main result of the paper states that the Banach space CΛBV of continuous functions of bounded Λ-variation with the standard Λ-variation norm is separable if and only if SΛ < 2. Also, ΛBV = ΛBVc if and only if SΛ < 2, where ΛBVc denotes the space of functions continuous in Λ-variation. A number of corollaries is drawn, and one of them being that the Garsia–Sawyer class GS is a dense subset of the Banach space HBV of functions of bounded harmonic variation. © 2008 Elsevier Inc. All rights reserved