The Continuous Functions on Normed Linear Spaces

The articles [16], [19], [20], [2], [21], [4], [9], [3], [1], [11], [15], [5], [17], [18], [10], [7], [8], [6], [13], [22], [12], and [14] provide the notation and terminology for this paper. We use the following convention: n is a natural number, x, X, X1 are sets, and s, r, p are real numbers. Let S, T be 1-sorted structures. A partial function from S to T is a partial function from the carrier of S to the carrier of T . For simplicity, we adopt the following rules: S, T denote real normed spaces, f , f1, f2 denote partial functions from S to T , s1 denotes a sequence of S, x0, x1, x2 denote points of S, and Y denotes a subset of S. Let R1 be a real linear space and let S1 be a sequence of R1. The functor −S1 yields a sequence of R1 and is defined as follows: (Def. 1) For every n holds (−S1)(n) = −S1(n). Next we state two propositions: (1) For all sequences s2, s3 of S holds s2 − s3 = s2 + −s3. (2) For every sequence s4 of S holds −s4 = (−1) · s4. Let us consider S, T and let f be a partial function from S to T . The functor ‖f‖ yielding a partial function from the carrier of S to R is defined as follows: (Def. 2) dom‖f‖ = dom f and for every point c of S such that c ∈ dom‖f‖ holds ‖f‖(c) = ‖fc‖.