A note on the uniform approximation of continuous affine functions

Abstract 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 c 0 and a continuous affine function on X , which does not belong to ( l 1 + R ) | X ¯ .