Linear Transformations of Euclidean Topological Spaces

For simplicity, we adopt the following rules: X, Y denote sets, n, m, k, i denote natural numbers, r denotes a real number, R denotes an element of RF, K denotes a field, f , f1, f2, g1, g2 denote finite sequences, r1, r2, r3 denote real-valued finite sequences, c1, c2 denote complex-valued finite sequences, and F denotes a function. Let us consider X, Y and let F be a positive yielding partial function from X to R. One can check that F Y is positive yielding. Let us consider X, Y and let F be a negative yielding partial function from X to R. One can verify that F Y is negative yielding. Let us consider X, Y and let F be a non-positive yielding partial function from X to R. Note that F Y is non-positive yielding.