Nearly exotic topologies on normed spaces

It is proved that for every infinite dimensional normed space (E, ‖ ‖) there is a non-trivial linear space topologyτ onE which is weaker than the norm topology and is such that (E, τ) admits no non-trivial continuous linear functionals. IfE is a space with a generalized basis or is aC(X) space, it is proved that the topologyτ can be taken to be Hausdorff.