On isomorphism of the space of continuous functions with finite p-th variation along a partition sequence

We study the concept of (generalized) p-th variation of a real-valued continuous function along a general class of refining sequence of partitions. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of ${\ell }^p$-norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to ${\ell }^{\infty }$-norm of the Schauder coefficients. This result provides an isomorphism between the space of α-Hölder continuous functions with finite (generalized) p-th variation along a given partition sequence and a subclass of infinite-dimensional matrices equipped with an appropriate norm, in the spirit of Ciesielski.