RESULT ON THE SPACE

Abstract

In this paper, we introduce a special space D(K,r,M,) of infinitely differentiable functions on K. We have proved Ro(K) CD(K,r,M,) if and only if (M) is a non-analytic sequence. We start with some standard definitions. Definition 1. Let K be a non-empty set, and let ƒ be a bounded complex-valued function on K. The uniform norm of ƒ on K, denoted by, is defined by = sup{f(x): xЄK). Definition 2. Let (X, d) be a metric space, and ACX. Let XE X. The distance between x and A, denoted by d (x, A), is defined by D(x, A) = inf {d (x, a) a € A}. Definition 3. Let L be a compact space. Then Ro(K) is the set of restriction functions to K of rational functions with poles off K. Notation. The set of all limits points of K is denoted by K'.