Is there a metrizable topology on $S’$ (the set of linear continuous functionals in Schwarz’s space) whose convergent sequences are the sequences that converge pointwise?
The obvious topology with these convergent sequences is the weak* topology, which I know is not metrizable. However, there might be another topology with the same convergent sequences which is metrizable.
I know that for smooth functions, there is no such metrizable topology whose convergent sequences are the pointwise convergent sequences.
But I don’t know how prove this. Maybe it is classical fact, but I didn’t find anything.