A Note on Convergence in Length

1. Introduction. Let J be a closed linear interval ao^tSfo. Let r(/) = (#(/), y(t) } z(t)), tSI, represent a vector function whose three components x(t), y(i) t z(t) are of bounded variation and continuous on I. This vector function determines in Euclidean 3-space a curve x~x{t), y~y(t), z=*z(t) whose length we denote by LQç). By convergence in length of a sequence of such vector functions $ n (t) = (x n (t), yn(t), Zn(t)), w = 0, 1, 2, • • • , is meant that x n (t), y nit), z n (t) converge uniformly on I to Xo(t), yo(t), Zo(t) respectively and that L($ n) converges to L(ïo). We denote by V(f) the total variation on 7 of a scalar function ƒ(t) which is continuous and of bounded variation on I. By convergence in variation of a sequence ƒ»(/), w = 0, 1, • • • , is meant that f n (t) is continuous and of bounded variation on I for w=*0, 1, • • • , that fn(t) converges uniformly on I to fo(t), and that V(fn)-+V(fo). These concepts are due to Adams, Clarkson, and Lewy [1, 2].