Recognizing locally convex spaces on which all bounded linear functionals are continuous 
Is it possible to characterize the Hausdorff locally convex spaces on which all bounded linear functionals are continuous?

It is known that a space is bornological if and only if the space is Mackey and every bounded linear functional is continuous. It follows that, in a way, I am interested in bornological spaces that are not Mackey, but this is not a usable characterization. (I am especially interested in the space of continuous functions on a Hausdorff completely regular space endowed with various topologies.)
 A: Your strict topology ($\beta_t$) is the strongest locally convex topology on $ C_b(T)$ coinciding  with the compact-open topology on the unit ball $ \{ x \in C_b(T): \sup_{t \in T} | x(t) |\leq 1 \}$. 
If each bounded linear functional on $ C_b(T)$
is $\beta_t$-continuous, then $T$ is compact. Suppose $T$ is not compact. Let $p\in\beta T\setminus T$ ($\beta T$ is the Stone-Čech compactification of $T$).  Define the subset $N$  of $C(\beta T)$ by  $N=\{f:0\le f\le 1, ~f(p)=1\}$. Labelling $N$ as $\{f_i : i\in I\}$ we make $I$ into a directed set by saying that $i\ge j$ iff $f_i\le f_j$. Let $g_i$ is the restriction of $f_i$ to $T$. Then the net $\{g_i : i\in I\}$ decreases pointwise to $0$. By the Dini theorem $g_i\rightarrow0$ uniformly on compact sets. By the above characterization of the strict topology we have that $g_i\rightarrow0$ in $\beta_t$. The linear functional $\delta_p(g)=f(p)$, where $f$ is the continuous extension of $g\in C_b(T)$ to $\beta T$, is bounded, so 
 $\delta_p(g_i)\rightarrow 0$. This is in contradiction with $\delta_p(g_i)=f_i(p)=1$.
