Relatively compact sets of Banach space-valued bounded-variation spaces

In this paper, we give criteria for the relatively compact sets of Banach space-valued bounded-variation spaces in the sense of Jordan and Banach space-valued bounded Wiener p-variation spaces as p is an element of (0, 1). Then, we give sufficient conditions for the relatively compact sets of others Banach space-valued bounded variation spaces, such as Banach space-valued bounded Wiener p-variation spaces as p is an element of (1, infinity), Banach space-valued bounded Wiener-Young variation spaces, Banach space-valued bounded Schramm variation spaces, Banach space-valued bounded Waterman variation spaces, Banach space-valued bounded Riesz variation spaces, and Banach space-valued bounded Korenblum variation spaces.