Discontinuous pointwise limit obstruction to weak compactness
= Discontinuous pointwise limit obstruction to weak compactness
A bounded sequence in the <space of continuous functions on a compact space> that has a discontinuous pointwise limit cannot lie in a weakly compact set. Compactness would give a weakly convergent subnet, and continuous point-evaluation functionals would identify its continuous limit with the prescribed pointwise limit. No assumption of sequential compactness is needed.