Uniform Continuity of Functions on Normed Complex Linear Spaces

For simplicity, we follow the rules: X, X1 denote sets, r, s denote real numbers, z denotes a complex number, R1 denotes a real normed space, and C1, C2, C3 denote complex normed spaces. Let X be a set, let C2, C3 be complex normed spaces, and let f be a partial function from C2 to C3. We say that f is uniformly continuous on X if and only if the conditions (Def. 1) are satisfied. (Def. 1)(i) X ⊆ dom f, and (ii) for every r such that 0 < r there exists s such that 0 < s and for all points x1, x2 of C2 such that x1 ∈ X and x2 ∈ X and ‖x1 −x2‖ < s holds ‖fx1 − fx2‖ < r. Let X be a set, let R1 be a real normed space, let C1 be a complex normed space, and let f be a partial function from C1 to R1. We say that f is uniformly continuous on X if and only if the conditions (Def. 2) are satisfied. (Def. 2)(i) X ⊆ dom f, and (ii) for every r such that 0 < r there exists s such that 0 < s and for all points x1, x2 of C1 such that x1 ∈ X and x2 ∈ X and ‖x1 −x2‖ < s holds ‖fx1 − fx2‖ < r.