Spaces of continuous and bounded functions over the field of geometric complex numbers

Following Grossman and Katz (Non-Newtonian Calculus, 1972), we construct the sets and of geometric complex-valued bounded and continuous functions, where A denotes the compact subset of the complex plane ℂ. We show that the sets and of complex-valued bounded and continuous functions form a vector space with respect to the addition and scalar multiplication in the sense of multiplicative calculus. Finally, we prove that and are complete metric spaces. MSC:26A06, 11U10, 08A05.