ON BOREL SETS IN FUNCTION SPACES WITH THE WEAK TOPOLOGY

Abstract

It is proved that the duality map $\langle\,,\rangle:(\ell^\infty,\hbox{weak})\times((\ell^\infty)^*,\hbox{weak}^* )\longrightarrow {\bf R}$ is not Borel. More generally, the evaluation $e:(C(K),\wk)\times K\longrightarrow{\bf R}$, $e(f,x) = f(x)$, is not Borel for any function space $C(K)$ on a compact $F$-space. It is also shown that a non-coincidence of norm-Borel and weak-Borel sets in a function space does not imply that the duality map is non-Borel.