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.