Continuous Functions on Real and Complex Normed Linear Spaces

The notation and terminology used here are introduced in the following papers: [25], [28], [29], [4], [30], [6], [14], [5], [2], [24], [10], [26], [27], [19], [15], [12], [13], [11], [31], [20], [3], [1], [16], [21], [17], [23], [7], [8], [22], [18], and [9]. For simplicity, we use the following convention: n denotes a natural number, r, s denote real numbers, z denotes a complex number, C1, C2, C3 denote complex normed spaces, and R1 denotes a real normed space. Let C4 be a complex linear space and let s1 be a sequence of C4. The functor −s1 yields a sequence of C4 and is defined by: (Def. 1) For every n holds (−s1)(n) = −s1(n). The following propositions are true: (1) For all sequences s2, s3 of C1 holds s2 − s3 = s2 + −s3. (2) For every sequence s1 of C1 holds −s1 = (−1C) · s1. Let us consider C2, C3 and let f be a partial function from C2 to C3. The functor ‖f‖ yielding a partial function from the carrier of C2 to R is defined by: (Def. 2) dom‖f‖ = dom f and for every point c of C2 such that c ∈ dom‖f‖ holds ‖f‖(c) = ‖fc‖. Let us consider C1, R1 and let f be a partial function from C1 to R1. The functor ‖f‖ yielding a partial function from the carrier of C1 to R is defined as follows: (Def. 3) dom‖f‖ = dom f and for every point c of C1 such that c ∈ dom‖f‖ holds ‖f‖(c) = ‖fc‖.