Assume $E$ is an infinite dimensional vector space. Assume $\tau_1, \tau_2$ are two topologies that make $E$ into topological vector spaces. Assume $\tau_1$ is strictly finer than $\tau_2$.
Is there any examples of $E, \tau_1, \tau_2$, such that the dual of $(E, \tau_1)$ is the same (as set of linear functions $E\to{\mathbb C}$) as the dual of $(E, \tau_2)$?
If $E$ is normed, and $\tau_1$ and $\tau_2$ are the strong and weak topologies, then $(E, \tau_1)$ and $(E, \tau_2)$ have the same duals, by definition: $\tau_2$ is the weakest/coarsest topology that makes the linear functionals on $E$ continuous. If $E$ is infinite-dimensional, then $\tau_1$ is strictly finer than $\tau_2$.