Partial Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

The notation and terminology used in this paper have been introduced in the following papers: [7], [15], [2], [3], [24], [4], [5], [1], [11], [16], [6], [9], [12], [17], [18], [10], [8], [23], [14], [21], [13], and [22]. For simplicity, we use the following convention: n, m denote non empty elements of N, i, j denote elements of N, f denotes a partial function from 〈Em, ‖ · ‖〉 to 〈En, ‖ · ‖〉, g denotes a partial function from Rm to Rn, h denotes a partial function from Rm to R, x denotes a point of 〈Em, ‖ · ‖〉, y denotes an element of Rm, and X denotes a set. We now state a number of propositions: