A bornologigal topological vector space is such that any bounded linear function on it is continuous. It is a standard result [Jarchow, Locally convex spaces, 1981] that if the dual $E'$ of a Mackey space $E$ is complete for the topology $\mathcal{T}\_{\mathcal{B}\_0}$ of uniform convergence on bipolars of null sequences, then the Mackey space is bornological. 


In a report thesis [Gach, Topological versus Bornological Concepts in Infinite Dimensions, 2004, Thm 6.1.16], this result even in stated for the strong topology on $E'$. Does this result appears anywhere else ? It seems quite strong to me, and I am not sure I understand the proof. 



Complete dual of bornological space
If I understand properly, I doubt very much that this is true.


I the article *On different types of non-distinguished Frechet spaes* (Note di Mat. 10 (1990), 149-165), Bonet, Dierolf, and Fernandez write that for a Frechet space $X$ with dual $(X',\beta(X',X))$ the Mackey topology $\mu(X',X'')$ is bornological if and only if every linear form on $X'$ which is bounded on bounded sets is already continuous.
Moreover, they show examples that this not always the case. It thus seems to me that $(E,\tau)=(X',\mu(X',X''))$ is a counterexample.


