There is no natural topology on duals of locally convex spaces (Q1327131)

The authors show that there is no ``natural topological duality'' on the category of locally convex spaces. That is, there does not exist any functor from the category of locally convex spaces into the category of topological spaces which associates to each \((X,\tau)\) a topology \(\tau\sphat\) on \((X,\tau)'\) such that the following two conditions hold: (1) For any pair of locally convex spaces \((X,\tau)\) and \((Y,\tilde\tau)\), a linear map \(f\colon X\to Y\) is \(\tau\)-\(\tilde\tau\)-continuous if and only if its linear adjoint \(f^*\) maps \((Y,\tilde\tau)'\) into \((X,\tau)'\) and the restriction of \(f^*\) to \((Y,\tilde \tau)'\) is \(\tilde \tau \sphat\)-\(\tau \sphat\)-continuous. (2) If \(\tau\) is the norm topology of any normed space \((X,\|\cdot\|)\), then \(\tau \sphat\) equals the topology induced by the dual norm\break \(\|\cdot\|^*\).