UNIFORM LIMITS OF SEQUENCES OF POLYNOMIALS AND THEIR DERIVATIVES

Let E be a compact subset of the unit interval [0, 1], and let C(E) denote the space of functions continuous on E with the uniform norm. Consider the densely defined operator D: C(E) --> C(E) given by Dp = p' for all polynomials p. Let G represent the graph of D, that is G = {(p, p'): p polynomials} considered as a submanifold of C(E) x C(E). Write the interior of the set E, int E as a countable union of disjoint open intervals and let E be the union of the closure of these intervals. The main result is that the closure of G is equal to the set of all functions (h, k) is-an-element-of C(E) x C(E) such that h is absolutely continuous on E and k\E = h'\E. As a consequence, the operator D is closable if and only if the set E is the closure of its interior. On the other extreme, G is dense in C(E) x C(E) i.e. for any pair (f, g) is-an-element-of C(E) x C(E), there exists a sequence of polynomials {p(n)} so that p(n) --> f and p(n)' --> g uniformly on E, if and only if the interior int E of E is empty.