Partial Differentiation, Differentiation and Continuity on n-Dimensional Real Normed Linear Spaces

The notation and terminology used in this paper have been introduced in the following papers: [2], [7], [1], [3], [4], [5], [17], [11], [13], [6], [9], [14], [10], [8], [12], and [18]. One can prove the following propositions: (1) Let n, i be elements of N, q be an element of Rn, and p be a point of En T. If i ∈ Seg n and q = p, then |pi| ≤ |q|. (2) For every real number x and for every element v1 of 〈E1, ‖ · ‖〉 such that v1 = 〈x〉 holds ‖v1‖ = |x|. (3) Let n be a non empty element of N, x be a point of 〈En, ‖ · ‖〉, and i be an element of N. If 1 ≤ i ≤ n, then ‖(Proj(i, n))(x)‖ ≤ ‖x‖.