The Differentiable Functions on Normed Linear Spaces

The notation and terminology used in this paper are introduced in the following papers: [20], [23], [4], [24], [6], [5], [19], [3], [10], [1], [18], [7], [21], [22], [11], [8], [9], [25], [13], [15], [16], [17], [12], [14], and [2]. For simplicity, we adopt the following rules: n, k denote natural numbers, x, X, Z denote sets, g, r denote real numbers, S denotes a real normed space, r1 denotes a sequence of real numbers, s1, s2 denote sequences of S, x0 denotes a point of S, and Y denotes a subset of S. Next we state several propositions: (1) For every point x0 of S and for all neighbourhoods N1, N2 of x0 there exists a neighbourhood N of x0 such that N ⊆ N1 and N ⊆ N2. (2) Let X be a subset of S. Suppose X is open. Let r be a point of S. If r ∈ X, then there exists a neighbourhood N of r such that N ⊆ X. (3) Let X be a subset of S. Suppose X is open. Let r be a point of S. If r ∈ X, then there exists g such that 0 < g and {y; y ranges over points of S: ‖y − r‖ < g} ⊆ X. (4) Let X be a subset of S. Suppose that for every point r of S such that r ∈ X there exists a neighbourhood N of r such that N ⊆ X. Then X is open. (5) Let X be a subset of S. Then for every point r of S such that r ∈ X there exists a neighbourhood N of r such that N ⊆ X if and only if X is open.