A note on the uniform approximation of continuous affine functions

It is known that if X is a compact convex subset of a locally convex space E, then the set of all continuous affine functions on X   equals the set (E*+R)|X¯. We study the possibility of extending this theorem for noncompact sets. For a bounded convex subset X of a locally convex space E   we characterize those continuous affine functions belonging to (E*+R)|X¯. We also give an example of a closed bounded convex subset X   of c0c0 and a continuous affine function on X  , which does not belong to (l1+R)|X¯.