Differentiable Functions into Real Normed Spaces

The notation and terminology used here have been introduced in the following papers: [12], [2], [3], [7], [9], [11], [1], [4], [10], [13], [6], [17], [18], [15], [8], [16], [19], and [5]. For simplicity, we adopt the following rules: F denotes a non trivial real normed space, G denotes a real normed space, X denotes a set, x, x0, r, p denote real numbers, n, k denote elements of N, Y denotes a subset of R, Z denotes an open subset of R, s1 denotes a sequence of real numbers, s2 denotes a sequence of G, f , f1, f2 denote partial functions from R to the carrier of F , h denotes a convergent to 0 sequence of real numbers, and c denotes a constant sequence of real numbers. We now state two propositions: (1) If for every n holds ‖s2(n)‖ ≤ s1(n) and s1 is convergent and lim s1 = 0, then s2 is convergent and lim s2 = 0G. (2) (s1 ↑ k) (s2 ↑ k) = (s1 s2) ↑ k. Let us consider F and let I1 be a partial function from R to the carrier of F . We say that I1 is rest-like if and only if: