When thinking about dual spaces and weak topologies, I came up with the following thought:
Let $(X,\tau)$ be a topological vector space and let $X^{\prime}$ be its (topological) dual space. Now, this dual space gives rise to a weak topology on $X$. Let us denote by $(X_w, \tau_w)$ the original space $X$ endowed with the weak topology. Now, I consider the dual space of this space, i.e. $X_w^{\prime}$. Then $X_w^{\prime} \subset X^{\prime}$. This dual space gives rise to an even weaker topology on $X$, which I denote by $(X_{ww},\tau_{ww})$ and so on and so forth. Iterating this, the topology becomes coarser and coarser. Can one say anything about the "limit topology"? Has this question been investigated at all? I haven't found anything about it in my search.
Thanks for your replies!
Best, Luke