On nonlinear metric spaces of functions of bounded variation

In the first part of the paper, the nonlinear metric space <(G) over bar (infinity)[ a, b], d > is defined and studied. It consists of functions defined on the interval [a, b] and taking the values in the extended numeric axis (R) over bar. For any x is an element of (G) over bar (infinity)[ a, b] and t is an element of (a, b) there are limit numbers x(t-0), x(t+0) is an element of(R) over bar (and numbers x(a+ 0), x(b- 0). (R) over bar). The completeness of the space is proved. It is the closure of the space of step functions in the metric d. In the second part of the work, the nonlinear space RL[a, b] is defined and studied. Every piecewise smooth function defined on [a, b] is contained in RL[a, b]. Every function x. RL[a, b] has bounded variation. All one-sided derivatives (with values in the metric space <(R) over bar,(sic)) are defined for it. The function of left-hand derivatives is continuous on the left, and the function of right-hand derivatives is continuous on the right. Both functions extended to the entire interval [a, b] belong to the space (G) over bar (infinity)[ a, b]. In the final part of the paper, two subspaces of the space RL[a, b] are defined and studied. In subspaces, promising formulations for the simplest variational problems are stated and discussed.