Some new results on sequence spaces with respect to non-Newtonian calculus

As an alternative to classical calculus, Grossman and Katz (Non-Newtonian Calculus, 1972) introduced the non-Newtonian calculus consisting of the branches of geometric, anageometric and bigeometric calculus etc. Following Grossman and Katz, we construct the field R(N) of non-Newtonian real numbers and the concept of non-Newtonian metric. Also, we give the triangle and Minkowski's inequalities in the sense of non-Newtonian calculus. Later, we respectively define the sets omega(N), l(infinity)(N), c(N), c(0)(N) and l(p)(N) of all, bounded, convergent, null and p-absolutely summable sequences in the sense of non-Newtonian calculus and show that each of the sets forms a vector space on the field R(N) and a complete metric space.